AFFINE RULINGS OF NORMAL RATIONAL SURFACES

DANIEL DAIGLE AND PETER RUSSELL

Given an algebraic surface X satisfying:

(†)

X is a complete normal rational surface, X is affine ruled and rank(Pic X_{s}) = 1,

where X_{s} denotes the smooth locus of X, consider: Problem 1. Find all affine rulings of X. Problem 2. Find all pairs of curves C_{1}, C_{2} on X such that X \(C_{1} ∪C_{2}) is isomorphic to ℙ^{2} minus two lines. Problem 3. Find all curves C in X such that κ̄(X_{s} \ C) = −∞.

This paper investigates Problem 1 for an arbitrary X satisfying (†). We define (1.14) the notion of a “basic” affine ruling of X and our main results describe how to construct all affine rulings of X, assuming that the basic ones are known. In the case where X is a weighted projective plane, the basic affine rulings of X are given in [6]; the present paper and [6] therefore constitute a solution to Problem 1 in that case.

Problem 3 (with X = ℙ^{2}) has been considered by several authors ([8], [9], [18], [19], [14]). In his review of [14] (see MR 82k:14013), M. H. Gizatullin mentions some unpublished examples found by V. I. Danilov and himself, and which seem to correspond to the list of basic affine rulings of ℙ^{2}. The case X = ℙ^{2} was finally solved in [10]. Our generalization to weighted projective planes seems to be new, as well as our method— valid for any X satisfying (†)—which reduces the general problem to the determination of the basic affine rulings.

Let us briefly indicate how problems 1–3 are related to each other. Consider the stronger condition (‡) on a surface X:

(‡)

X satisfies (†) and every singular point of X is a cyclic quotient singularity.

As an example, note that the weighted projective planes satisfy (‡) (they even satisfy Pic(X_{s}) = ℤ; see [6] for these claims). Also note the following by-product of section 1: A surface satisfying (‡) cannot have more than 3 singular points (see Corollary 1.16).

1991 Mathematics Subject Classification. Primary: 14J26. Secondary: 14J17, 13B10.

Research supported by CICMA and grants from NSERC Canada.

It is clear that any solution (C_{1}, C_{2}) to Problem 2 gives rise to an affine ruling of X; by Theorem 1.15, the converse holds if X satisfies (‡), so:

For any surface X satisfying (‡), Problems 1 and 2 are equivalent.

The exact relation between Problem 3 and the other two is given by the following statement, which will be proved in 1.17, below: Given X satisfying (‡) and a curve C on X, the following are equivalent:

(i) κ̄(X_{s} \ C) = −∞; (ii) there exists at least one affine ruling^{1} Λ of X such that nC ∈ Λ for some n > 0. For instance, if C ⊂ ℙ^{2} is Yoshihara’s rational quintic ([19], Proposition 3, case

N = 1), then infinitely many affine rulings Λ of ℙ^{2} contain multiples of C.

By way of motivation, we now explain the connection between problems 1–3 and locally nilpotent derivations. Consider the polynomial ring B = k[X_{1}, X_{2}, X_{3}], where k is an algebraically closed field of characteristic zero. It is known ([13], [3]) that describing the locally nilpotent derivations D : B → B is equivalent to answering: Which pairs of polynomials f, g ∈ B have the property that k[f, g] is the kernel of a locally nilpotent derivation of B? If we restrict ourselves to the case where D is (or equivalently f and g are) homogeneous with respect to weights w(X_{i}) = a_{i}, where a_{1}, a_{2}, a_{3} are relatively prime positive integers, then we can think of f and g as defining curves in the weighted projective plane ℙ(a_{1}, a_{2}, a_{3}) = Proj B; then [4] gives the following result: Theorem. For w-homogeneous elements f, g ∈ B satisfying gcd(w(f), w(g)) = 1, the following are equivalent:

(1) There exists a w-homogeneous locally nilpotent derivation D of B such that ker D = k[f, g];

(2) f and g are irreducible elements of B and the algebraic surface Proj B \ V (fg) is isomorphic to ℙ^{2} minus two lines.

Note that the case where gcd(w(f), w(g)) ≠ 1 turns out to be very special, and is completely described in [4]. Hence, solving Problem 2 for X = ℙ(a_{1}, a_{2}, a_{3}) is equivalent to describing homogeneous locally nilpotent derivations of B. Since that class of derivations is not well understood, and corresponds to a class of G_{a}-actions on 𝔸^{3} which ought to be understood, there is ample reason to study affine rulings.

Organization of the text Fix a surface X satisfying (†). Section 1 contains generalities about affine rulings of X. Section 2 defines a process which is used to modify affine rulings of X (i.e., applying

it to an affine ruling of X produces a different affine ruling of X). The process makes its first appearance in the proof of Theorem 2.1, where it is shown that every non-basic affine ruling of X can be “reduced” to a simpler one; this reduction process is in fact a special case of the modification process.

1According to the definition of “affine ruling” adopted in 1.1, below, Λ is a linear system of X, so it makes sense to write nC ∈ Λ.

Some preparation is necessary before defining the modification process: 2.2 defines the notion of an “X-immersion”; then 2.3–2.8 show that each X-immersion determines an affine ruling of X, that each affine ruling can be obtained in this way, and that this can be turned into a bijective correspondence, modulo appropriate adjustments.

Given an X-immersion I, 2.9 defines a set Π(I) and a new X-immersion I ∗π for each π ∈ Π(I). This operation ∗ is the modification process which was announced; it acts on X-immersions, so it indirectly modifies affine rulings via the correspondence mentioned in the preceding paragraph. Discussion 2.14 summarizes the results of section 2. In particular, it states that all affine rulings of X can be constructed from the basic ones by using the ∗ operation; and consequently the solution of Problem 1 consists of two parts:

(1a) Make a list of all basic affine rulings of X.

(1b) For each X-immersion I, describe the set Π(I).

Problem (1b) is essentially a problem in the theory of weighted graphs, independent of the surface, and is completely solved in sections 3 and 4: section 3 does the graph theory and section 4 states the consequences for Π(I). This paper does not solve Problem (1a), which is highly dependent on the surface X; [6] solves it for the weighted projective planes.

In contrast with sections 2 and 4, where rulings are described by saying that they can be constructed from basic ones by using the modification process, section 5 gives direct information on affine rulings. The main result of that section is Theorem 5.13; it is complemented by several other (more practical) statements, notably 5.17, 5.22, 5.23, 5.25, 5.34, 5.40.

Conventions

All curves and surfaces considered in this paper are assumed to be algebraic varieties over an algebraically closed field k of characteristic zero. In particular, curves and surfaces are irreducible and reduced.

If f : X → Y is a birational morphism of surfaces then the center of f (denoted center(f)) is the set of points y ∈ Y such that f^{−1}(y) contains more than one point.

Let S be a smooth complete surface. If D is a divisor of S then, by a component of D, we always mean an irreducible (or prime) component of D. If D and D′ are divisors of S then D · D′ denotes their intersection number and D^{2} = D · D. If C ⊂ S is a smooth rational curve and C^{2} = r, we call C an r-curve; by an r-component of a divisor D, we mean a component of D which is an r-curve. A reduced effective divisor D of S has strong normal crossings if: (i) each component of D is a smooth curve; (ii) if D_{i} and D_{j} are distinct components of D then D_{i} · D_{j} ≤ 1; and (iii) if D_{i}, D_{j} and D_{k} are distinct components of D then D_{i} ∩ D_{j} ∩ D_{k} is empty.

Except for the graph 𝕃(X) of 2.14, every graph considered in this paper is a weighted graph, i.e., a graph in which each vertex is assigned an integer (called its weight). Every weighted graph in this paper is a finite undirected graph such that no edge connects a vertex to itself and at most one edge joins any given pair of vertices.

If S is a smooth complete surface and D a divisor of S with strong normal crossings, the dual graph of (D, S) is the weighted graph G = G(D, S) whose vertices are the components of D; distinct vertices D_{i} and D_{j} are joined by an edge if D_{i} ∩D_{j} ≠ ∅; and the weight of a vertex D_{i} is D_{i}^{2}. We assume familiarity with this idea, as well as with the basic theory of weighted graphs (their blowing-up, blowing-down and equivalence); the relevant definitions can be found in various sources, for instance [17], [16], or the appendix of [2] (see also the beginning of section 3, in this paper). Let D_{1}, …, D_{n} be the distinct components of D. We say that D_{j} is a neighbor of D_{i} if i ≠ j and D_{i} ∩ D_{j} ≠ ∅ (i.e., if the vertices D_{i}, D_{j} of G are neighbors); the number of neighbors of D_{i} is called its branching number; if this number is greater than or equal to 3, we say that D_{i} is a branching component of D (or that the vertex D_{i} is a branch point of G). We say that G is a linear chain (or a linear tree) if it is a tree without branch points; an admissible chain is a linear chain in which every weight is strictly less than −1; note that the empty graph is an admissible chain. We say that D is a tree (or a linear chain, or an admissible chain, etc) if G has the corresponding property.

Let X and X^{∗} be complete normal surfaces, β a birational isomorphism between them (either X →^{β} X^{∗} or X ←^{β} X^{∗}) and Λ a one-dimensional linear system on X without fixed components. In this situation, we will often use the fact that Λ and β determine, in a natural way, a one-dimensional linear system Λ^{∗} on X^{∗} without fixed components. The tacit understanding is that, for suitably chosen rational maps X →^{λ} ℙ^{1} and X^{∗} →^{λ^{∗}} ℙ^{1} determining Λ and Λ^{∗} respectively, β, λ and λ^{∗} form a commutative diagram.

The set of nonnegative (resp. positive) integers is denoted ℕ (resp. ℤ^{+}).

1. Preliminaries on affine rulings

1.1. Let X be a complete normal rational surface. An “affine ruling” of X is usually defined to be a morphism p : U → Γ where Γ is a curve, U is a nonempty open subset of X isomorphic to Γ × 𝔸^{1} and p is the projection Γ × 𝔸^{1} → Γ. Since Γ × 𝔸^{1} is normal and rational, Γ is an open subset of ℙ^{1} and U is contained in the smooth locus of X. The morphism p extends to a rational map X → ℙ^{1} which, in turn, determines a unique linear system Λ on X without fixed components. Since we do not want to distinguish between rulings which determine the same linear system Λ, we adopt the viewpoint that Λ itself is the affine ruling: Definition. Let Λ be a one-dimensional linear system on X without fixed components. We say that Λ is an affine ruling of X if there exist nonempty open subsets U ⊂ X and Γ ⊆ ℙ^{1} such that U ≅ Γ × 𝔸^{1} and such that the projection morphism Γ × 𝔸^{1} → Γ determines Λ.

If Λ is an affine ruling of X then the general member C of Λ satisfies C ∩ U ≅ 𝔸^{1}; it follows:

• the general member of Λ is irreducible and reduced;

• Λ has at most one base point on X.

In the special case where X is smooth and Bs(Λ) = ∅, the general member C of Λ satisfies C ≅ ℙ^{1} and C^{2} = 0; so 1.2 applies to this situation.

1.2. Let X be a smooth, complete rational surface and C a curve on X satisfying C ≅ ℙ^{1} and C^{2} = 0. Then the following facts are well-known (see 2.7.1 of [11] or Lemma 2.2 of [12], p. 115):

(1) The Riemann-Roch Theorem for X implies that the complete linear system Λ = |C| has dimension one; since Bs(Λ) = ∅, Λ gives rise to a morphism λ : X → ℙ^{1}.

(2) There exists an open subset Γ ≠ ∅ of ℙ^{1} such that λ^{−1}(Γ) ≅ Γ × ℙ^{1} and such that the composition λ^{−1}(Γ) ≅ Γ × ℙ^{1} → Γ is the restriction of λ (i.e., Λ is a ℙ^{1}-ruling of X).

(3) There exists an irreducible curve H ⊂ X such that H · Λ = 1; such a curve H is called a section of Λ (or λ). If H is a section then H ≅ ℙ^{1} and, given Γ satisfying (2) and Γ ≠ ℙ^{1}, we have λ^{−1}(Γ) \ H ≅ Γ × 𝔸^{1} and the composition λ^{−1}(Γ) \ H ≅ Γ × 𝔸^{1} → Γ is the restriction of λ (so Λ is also an affine ruling of X).

(4) If U is any open subset of X isomorphic to Γ × 𝔸^{1} for some open subset Γ ≠ ∅ of ℙ^{1}, and if the composition U ≅ Γ × 𝔸^{1} → Γ is compatible with Λ, then U = X \ supp(H + C_{1} + ⋯ + C_{r}) for some section H of Λ and for some curves C_{1}, …, C_{r} where each C_{i} is contained in some member of Λ.

Let H be a section of Λ, let m = −H^{2} and, for each reducible^{2} member F of Λ, let F^{∘} be the unique irreducible component of F which meets H (F^{∘} is an integral curve and occurs in F with multiplicity one).

(5) For each reducible member F of Λ, if F^{♯} denotes the reduced effective divisor such that supp(F) = supp(F^{♯}) then F^{♯} has strong normal crossings and is a tree of projective lines. Moreover, F^{♯} can be shrunk until only F^{∘} remains (F^{∘} itself is not shrunk) and, after that contraction, (F^{∘})^{2} = 0. Note the following consequence: if C is an irreducible component of F and C is branching in supp(F + H) then C^{2} < −1.

(6) If all reducible members are shrunk as described in (5), then one obtains the ruled surface 𝔽_{m}. This shrinking process is a birational morphism σ : X → 𝔽_{m} which maps the members of Λ (resp. H) to the members (resp. the negative section) of the ruling of 𝔽_{m}.

2Note that if F ∈ Λ has irreducible support then the condition F · H = 1 implies that it is also reduced (i.e., F is an integral curve).

The shrinking processes described in (5) and (6) are uniquely determined by the choice of a section H. Notation 1.3. If X is a complete normal rational surface and Λ is an affine ruling of X, let X_{s} be the smooth locus of X and X′ = X_{s} \ Bs(Λ). We write (X̄, Λ̄) ≻ (X, Λ) to indicate that X̄ is a smooth and complete surface containing X′ as an open subset, the complement of X′ in X̄ is the support of a reduced effective divisor with strong normal crossings, Λ̄ is a base point free affine ruling of X̄ and Λ̄|_{X′} is equal to Λ|_{X′}. Lemma 1.4. Let X be a complete normal rational surface and Λ an affine ruling of X and suppose that (X̄, Λ̄) ≻ (X, Λ). Let D be the divisor of X̄ with strong normal crossings and whose complement is X′. Then:

(1) Each connected component of D is a tree of projective lines.

(2) At most one irreducible component of D is a section of Λ̄.

(3) Every irreducible component of D which is not a section of Λ̄ is contained in a member of Λ̄.

Proof. Consider an open subset U ⊂ X isomorphic to Γ × 𝔸^{1} (for some open subset Γ ≠ ∅ of ℙ^{1}) and such that the composition U ≅ Γ × 𝔸^{1} → Γ is compatible with Λ; note that U ⊆ X′. Since the complement of Γ × 𝔸^{1} in ℙ^{1} × ℙ^{1} is a tree of projective lines, and since supp(D) is contained in X̄ \U, it easily follows that assertion (1) holds. By part (4) of 1.2 we have X̄ \ U = supp(H + C_{1} + ⋯ + C_{r}), for some section H of Λ̄ and for some curves C_{1}, …, C_{r}, where each C_{i} is contained in some member of Λ̄; since supp(D) ⊆ supp(H + C_{1} + ⋯ + C_{r}), (2) and (3) hold. □

Proposition 1.5. Let X be a complete normal rational surface and Λ an affine ruling of X. Let X′ = X_{s} \ Bs(Λ).

(1) There exists a unique pair (X, Λ)^{∼} = (X̃, Λ̃) satisfying (X̃, Λ̃) ≻ (X, Λ) and the following condition: (*) Every irreducible component C of X̃ \ X′ satisfies C^{2} ≤ −1, and if equality

holds then C is a section of Λ̃.

(2) Every irreducible component of X̃ \ X′ which is not a section of Λ̃ is contained in some reducible member of Λ̃.

(3) Every member of Λ̃ meets X′.

(4) If (X̄, Λ̄) is any pair satisfying (X̄, Λ̄) ≻ (X, Λ), then there exists a birational morphism X̄ → X̃ which restricts to an isomorphism from X′ ⊂ X̄ to X′ ⊂ X̃.

Proof. We begin by proving (4): assume that (X̃, Λ̃) is any pair satisfying (X̃, Λ̃) ≻ (X, Λ) and condition (*), and let (X̄, Λ̄) be as in assertion (4). There exists a smooth complete surface S and two birational morphisms, π̃ : S → X̃ and π̄ : S → X̄, such that if we regard π̃ (resp. π̄) as a composition of monoidal transformations then each one of these is centered at a point infinitely near X̃ \X′ (resp. X̄ \X′). We also assume that (S, π̃, π̄) is minimal, i.e., that the total number of monoidal transformations in π̃ and π̄ is minimal. It suffices to show that π̄ is an isomorphism.

Assume that π̄ is not an isomorphism and consider a curve Γ ⊂ S which is first to shrink, in the contraction process going from S to X̄. By minimality of (S, π̃, π̄), Γ is not in the exceptional locus of π̃; thus it is the strict transform of some component C of X̃ \ X′, where C^{2} ≥ −1. Since (X̃, Λ̃) satisfies (*), C must be a section of Λ̃. Since π̄(Γ) is a point, it follows that Λ̄ has a base point, contradicting (X̄, Λ̄) ≻ (X, Λ). Hence, π̄ is an isomorphism and (4) is proved.

Note that (4) implies, in particular, that if (X̃, Λ̃) exists then it is unique (up to isomorphism). So, to finish the proof, there remains to construct a pair (X̃, Λ̃) satisfying (1–3).

Consider the minimal resolution of singularities X̂ → X of X and let Ê be the inverse image of the singular points. Then X̂ is a smooth complete surface, Ê is a reduced effective divisor of X̂ with strong normal crossings and X̂ \ supp(Ê) → X_{s} is an isomorphism. Arguing as in the proof of 1.4, we see that each connected component of Ê is a tree of projective lines. Moreover, every irreducible component E of Ê satisfies E^{2} ≤ −1, and if E^{2} = −1 then E is branching in Ê.

Then Λ determines an affine ruling Λ̂ of X̂. Let ρ : X̃ → X̂ be the minimal resolution of the base points of Λ̂ and Λ̃ the corresponding base point free linear system on X̃. It is clear that (X̃, Λ̃) ≻ (X, Λ); we shall now argue that (*) holds. Let D be the divisor of X̃ with strong normal crossings such that X̃ \ X′ = supp(D) and consider a component C of D.

Since D is the union of the strict transform of Ê and of the exceptional locus of ρ, it is clear that C^{2} ≤ −1.

Assume that C is not a section of Λ̃. Then Lemma 1.4 implies that C is contained in some member F of Λ̃; since F^{2} = 0 and C^{2} < 0, F must have reducible support, which proves assertion (2) of the Proposition. There remains to show that C^{2} < −1. Assume the contrary; then C^{2} = −1 and, by 1.2, C is not branching in supp(H + F) for any section H of Λ̃.

Suppose that C is the strict transform of some component E of Ê. Then E^{2} ≥ −1 in X̂; by the properties of Ê, E^{2} = −1 and E is branching in Ê. Consider three distinct neighbors E_{i} (i = 1, 2, 3) of E in Ê. Since E^{2} = C^{2}, we see that the strict transform C_{i} of E_{i} meets C in X̃ (for all i = 1, 2, 3). Since C_{i} is a component of X̃ \X′, Lemma 1.4 implies that C_{i} is a section of Λ̃ or is contained in some member F_{i} of Λ̃; in the latter case, C_{i} ∩ C ≠ ∅ implies that F_{i} = F. Since at most one C_{i} can be a section of Λ̃ it follows that, for a suitable section H, C_{1}, C_{2}, C_{3} and C are all contained in supp(H + F). This contradicts the fact that C is not branching in supp(H + F), so C is not the strict transform of a component of Ê.

Thus C is in the exceptional locus of ρ (and Λ̂ has a base point). Write ρ = ρ_{r}∘⋯∘ρ_{1}, where ρ_{i} : X_{i} → X_{i−1} is a monoidal transformation (r ≥ 1, X_{0} = X̂, X_{r} = X̃), and note that the exceptional curve H ⊂ X̃ of ρ_{r} is a section of Λ̃. By (1.1), there is a unique base point on X_{i−1} (1 ≤ i ≤ r); it follows that the center of ρ_{i} lies on the exceptional curve of ρ_{i−1} for each i > 1, and consequently C^{2} = −1 implies C = H. This contradicts our assumption that C is not a section of Λ̃, so we proved that C^{2} < −1.

To prove (3), suppose that F ∈ Λ̃ satisfies supp(F) ⊆ supp(D). Then each component C of F satisfies C^{2} < −1 because C ⊆ supp(D) and C is not a section. This contradicts the fact (1.2) that F contracts to a 0-curve (or is a 0-curve). □

Definition 1.6. Suppose that X is a complete normal rational surface and that Λ is an affine ruling of X. Let X′ = X_{s} \ Bs(Λ) and consider (X, Λ)^{∼} = (X̃, Λ̃).

For each member F of Λ, let F̃ be the unique element of Λ̃ such that F̃∩X′ = F ∩X′; then F ↦ F̃ defines a bijection Λ → Λ̃ (because Λ̃|_{X′} = Λ|_{X′} and, by 1.5, each member of Λ̃ meets X′). 1.7. Let X be a complete normal rational surface and Λ an affine ruling of X. In this paragraph, we relate the rank of Pic(X_{s}) to some numbers determined by the pair (X̃, Λ̃) of Proposition 1.5.

Let D be the divisor of X̃ with strong normal crossings such that X̃ \X′ = supp(D). Proposition 1.5 implies that, for a suitable choice of a section H of Λ̃, every component C of D satisfies

(i) C^{2} ≤ −1 and one of

(ii) C = H (iii) C is contained in some reducible member of Λ̃ and C^{2} < −1. Let m = −H^{2} and let F_{1}, …, F_{s} be the reducible members of Λ̃. For each i, we

can write F_{i} = F_{i}^{∘} + F_{i}^{⋆} where F_{i}^{∘} is an integral curve, F_{i}^{∘} · H = 1, F_{i}^{⋆} is effective and F_{i}^{⋆} · H = 0. By 1.2, F_{i}^{⋆} can be shrunk to a point and, if we do this for all i = 1, …, s, we obtain the ruled surface 𝔽_{m}. Since Pic(𝔽_{m}) is freely generated by a section and a fibre, it follows that

(1)

Pic(X̃) is freely generated by H, a general member F of Λ̃ and all components of F_{1}^{⋆}, …, F_{s}^{⋆}.

We write F_{i}^{⋆} = F′_{i} + F′′_{i}, where F′_{i} and F′′_{i} are effective, F′_{i} contains the components of F_{i}^{⋆} which meet X′ and F′′_{i} contains those included in X̃ \ X′. We claim that F′_{i} ≠ 0. In fact, consider a component C of F_{i}^{⋆} satisfying C^{2} = −1 (such a C exists, since F_{i}^{⋆} is nonzero and shrinks to a point). Since C satisfies neither (ii) nor (iii), it is not a component of D, so C is contained in F′_{i}. Hence, F′_{i} ≠ 0 for all i.

Observe that

(2)  X̃ \ X′ = supp (δH + ∑_{i=1}^{s}(F′′_{i} + δ_{i}F_{i}^{∘})) ,

where

δ = \begin{cases}1, & if H ∩ X′ = ∅, \\ 0, & if H ∩ X′ ≠ ∅, \end{cases} and δ_{i} = \begin{cases}1, & if F_{i}^{∘} ∩ X′ = ∅, \\ 0, & if F_{i}^{∘} ∩ X′ ≠ ∅. \end{cases}

In view of (1), (2) and the fact that, for each i, F is linearly equivalent to F_{i} = F_{i}^{∘} + F′_{i} + F′′_{i}, we obtain that Pic(X′) is the abelian group generated by H, F and all components of F′_{1}, …, F′_{s}, with relations:

(3)

F = F′_{i} (for each i such that δ_{i} = 1) and
H = 0 (if δ = 1).

Note that Pic(X_{s}) = Pic(X′) and let k_{i} ≥ 1 be the number of components of F′_{i}. We conclude that

(4)  rank(Pic X_{s}) = (2 − δ) + ∑_{i=1}^{s}(k_{i} − δ_{i}),

where 1 ≤ 2 − δ ≤ 2 and, for all i, k_{i} − δ_{i} ≥ 0.

Surfaces satisfying the condition (†)

From now-on, we restrict ourselves to the case where X satisfies the condition (†) defined in the introduction. Proposition 1.8. Suppose that X satisfies (†), let Λ be an affine ruling of X and consider the pair (X̃, Λ̃) = (X, Λ)^{∼}.

(1) Λ has one base point on X and exactly one irreducible component H of X̃ \ X′ is a section of Λ̃.

(2) Every member F of Λ̃ has a unique irreducible component C_{F} which meets X′. Consequently, every member of Λ has irreducible support.

(3) If F is reducible then C_{F}^{2} = −1 and C_{F} is the only component of F with this property. Moreover, C_{F} does not meet H, is not branching in supp(F + H) and the multiplicity of C_{F} in F is strictly greater than 1.

(4) Under the bijection Λ → Λ̃ defined in 1.6, the multiple members of Λ correspond to the reducible members of Λ̃. If M = νC ∈ Λ, where C ⊂ X is a curve and ν ∈ ℕ, then ν is equal to the multiplicity of CM̃ in M̃.

(5) Let M_{i} = ν_{i}C_{i} (1 ≤ i ≤ s) be the multiple members of Λ, where C_{i} ⊂ X is a curve and ν_{i} > 1 is an integer, and let M be any member of Λ. Then Pic(X_{s}) is the abelian group given by s+1 generators M, C_{1}, …, C_{s} and relations ν_{i}C_{i} = M for i = 1, …, s. In particular, Pic(X_{s}) = ℤ if and only if s < 2 or ν_{1}, …, ν_{s} are pairwise relatively prime.

Proof. Let H be a section of Λ̃ satisfying conditions (i–iii) of 1.7; let the notations F_{i}, F_{i}^{∘}, F_{i}^{⋆}, F′_{i} and F′′_{i} be as in 1.7.

We have 1 = (2 − δ) + ∑_{i=1}^{s}(k_{i} − δ_{i}) by equation (4), where 2 − δ ≥ 1 and k_{i} − δ_{i} ≥ 0 for all i; thus δ = 1 and k_{i} = 1 = δ_{i} for all i = 1, …, s. Since δ = 1, H ∩ X′ = ∅ and assertion (1) is proved.

Let F ∈ Λ̃. If F is irreducible then F^{2} = 0 implies that F ∩ X′ ≠ ∅, by condition (i) of 1.7. If F = F_{i} for some i, then F_{i}^{∘} ∩ X′ = ∅ (because δ_{i} = 1) and F′_{i} has irreducible support (because k_{i} = 1). Assertion (2) follows.

We have F′_{i} = ν_{i}C_{F_{i}} for some ν_{i} ≥ 1. In 1.7, when we proved that F′_{i} ≠ 0, we actually showed that at least one component C of F′_{i} satisfies C^{2} = −1; thus C_{F_{i}}^{2} = −1. Conversely, if C is any component of F_{i} such that C^{2} = −1, then C ∩X′ ≠ ∅ (otherwise conditions (i–iii) of 1.7 would be violated), so C = C_{F_{i}}. Since F′_{i} does not meet H, C_{F_{i}} does not meet H; C_{F_{i}} is not branching in supp(F + H) because, in the contraction of F_{i} to a 0-curve, C_{F_{i}} is the first component to shrink. By part (6) of Lemma 2.2 of [12], C_{F_{i}} must be a multiple component of F_{i}. So (3) holds.

In part (4), the assertion about ν is trivial and the correspondence between multiple members of Λ and reducible members of Λ̃ is essentially the fact that C_{F_{i}} is a multiple component of F_{i} (preceding paragraph).

Since δ = 1 and δ_{i} = 1 for all i, and in view of (3) of 1.7, Pic(X_{s}) is generated by F, C_{F_{1}}, …, C_{F_{s}}, with relations ν_{i}C_{F_{i}} = F for i = 1, …, s. This, together with (4), implies (5). □

1.9. Suppose that X satisfies (†), let Λ be an affine ruling of X and consider the pair (X̃, Λ̃) = (X, Λ)^{∼}. By 1.2, each reducible member of Λ̃ can be shrunk to a 0-curve and the shrinking is uniquely determined by the choice of a section of Λ̃. From now-on, whenever we shrink reducible members of Λ̃ to 0-curves, we tacitely assume that the shrinking is the one which is determined by the unique section of Λ̃ contained in X̃ \X′ (see Proposition 1.8). 1.10. The following notations and remarks are useful. Suppose that X satisfies (†), let Λ be an affine ruling of X, consider (X̃, Λ̃) = (X, Λ)^{∼} and let D be the divisor of X̃ with strong normal crossings such that X̃ \ X′ = supp(D).

By Proposition 1.8, Λ̃ has a unique section H contained in D and, if F is a reducible member of Λ̃, F has a unique component C_{F} which meets X′ and the multiplicity ν of C_{F} in F is strictly greater than 1; moreover, supp(F − νC_{F}) has either one or two connected components and exactly one of those components meets H. Let us denote those connected components by F^{u} and F^{ℓ}, where F^{u} is the one which meets H and F^{ℓ} is allowed to be empty. We regard F^{u} and F^{ℓ} either as sets or as reduced effective divisors; we have F^{u} ≠ ∅ and, recalling how the morphism X̃ → 𝔽_{m} contracts F (see 1.2, 1.9), we see that F^{ℓ} is either empty or an admissible chain. Finally, let D_{0} denote the connected component of D which contains H; thus D_{0} = H + F_{1}^{u} + ⋯ + F_{s}^{u}, where F_{1}, …, F_{s} are the reducible members of Λ̃, and D = D_{0} + F_{1}^{ℓ} + ⋯ + F_{s}^{ℓ}.

As explained in 1.1, our definition of “affine ruling” is slightly different from the standard one. The following gives the exact relation between the two definitions: Proposition 1.11. Suppose that X satisfies (†) and that Λ is an affine ruling of X. For an open subset U of X, the following are equivalent:

(1) There exists an isomorphism U ≅ Γ × 𝔸^{1}, for some open subset Γ ≠ ∅ of ℙ^{1}, such that the composition U ≅ Γ × 𝔸^{1} → Γ is compatible with Λ.

(2) U = X \ supp(M_{1} + ⋯ + M_{p}), for some nonempty subset {M_{1}, …, M_{p}} of Λ containing in particular all multiple members.

Moreover, if these conditions hold (and M_{1}, …, M_{p} are distinct) then U is isomorphic to (ℙ^{1} − p points) × 𝔸^{1} (or equivalently to ℙ^{2} minus p lines meeting at a point).

Some graph theory is needed for proving the above result. Given q ∈ ℕ, let S_{q} be the weighted tree consisting of q + 1 vertices v_{0}, v_{1}, …, v_{q}, all of weight 0, and of the q edges {v_{0}, v_{i}}, i = 1, …, q. Note that det(S_{1}) = −1 and that det(S_{q}) = 0 for all q ≠ 1 (see 3.15 for the determinant of a weighted graph). Note that if q ≥ 1 and S is identical to S_{q} except for the weight of v_{0}, then S is equivalent to S_{q}. Note, also, that if S_{p} and S_{q} are equivalent then p = q. Lemma 1.12. Let p ≥ 1 and r ≥ 0 be integers, G a weighted tree, v a vertex of G, A_{1}, …, A_{p}, B_{1}, …, B_{r} the branches of G at v, where each A_{i} consists of a single vertex of weight 0 and, in each B_{i}, every weight is strictly less than −1.

(1) If G is equivalent to S_{q} for some q ∈ ℕ, then r = 0 and p = q.

(2) If G is equivalent to a linear chain Γ of the form

(∗)  [figure: …; 0; x; ω_{1}; ω_{q}] (q ≥ 0, ω_{i} ≤ −2 and x ∈ ℤ),

then G itself has the form (∗).

Proof. Let us say, temporarily, that a weighted tree T satisfies the condition (NN) if it has a branch point b such that: (i) at least one branch of T at b has all its weights strictly less than −1; and (ii) every branch of T at b containing a weight ≥ −1 contains a nonnegative weight. Then we leave it to the reader to verify the following fact:

(5)

If a weighted tree T satisfies (NN) then so does every minimal weighted tree equivalent to T .

Note that S_{q} is minimal and does not satisfy (NN); also, Γ contracts to a minimal chain which does not satisfy (NN). Since G is equivalent to S_{q} or Γ, it follows from (5) that G does not satisfy (NN). We claim:

(6)  If r ≠ 0 then G is of the form (∗) and det(G) ≤ −2.

Indeed, suppose that r ≠ 0; if either v or some vertex of some B_{i} is a branch point of G, then G satisfies (NN), a contradiction. So G is a linear chain. In particular, p + r ≤ 2, so p = 1 = r and G is of the form (∗). We have det(G) = − det(B_{1}) by 3.18, and det(B_{1}) ≥ 2 by 3.19; so (6) holds.

To prove assertion (1), suppose that G is equivalent to S_{q}. Then det(G) = det(S_{q}) ≥ −1, so r = 0 by (6). Since r = 0 and p > 0, G is equivalent to S_{p}, so p = q.

To prove (2), suppose that G is equivalent to Γ. By (6), we may assume that r = 0. Then G is equivalent to S_{p}, so det(S_{p}) = det(Γ) = − det(Γ′) ≤ −1 by 3.18 and 3.19, where Γ′ is the admissible chain with weights ω_{1}, …, ω_{q}. So p = 1 (and r = 0) and consequently G is of the form (∗) (with q = 0). □

Proof of Proposition 1.11. We shall prove that (1) implies (2) and leave the rest to the reader. Suppose that U satisfies condition (1) and let q ∈ ℕ be such that Γ = ℙ^{1} − q points. Regard U as an open subset of ℙ^{1} × ℙ^{1}; then the complement of U is a divisor W with strong normal crossings and whose dual graph is S_{q}. Note that U is connected at infinity. We also observe that U ⊂ X′, where X′ = X_{s} \ Bs(Λ); the inclusion is strict because the complement of U in X has pure dimension one (the intersection matrix of W is not negative definite, so W cannot be shrunk to a normal point).

Consider (X̃, Λ̃) = (X, Λ)^{∼} and recall that the open subset X′ of X can be embedded in X̃ as the complement of a divisor D of X̃ with strong normal crossings. Since U ⊂ X′ (strictly) and X̃ \ U has pure dimension one,

(7)  X̃ \ U = supp(D + C_{1} + ⋯ + C_{p}) (p > 0)

for some distinct curves C_{1}, …, C_{p} not contained in D. By Proposition 1.8, some component H of D is a section of Λ̃; thus part 4 of 1.2 implies that each C_{i} is contained in a member of Λ̃. Since every member F of Λ̃ has a unique component C_{F} which meets X′ (1.8), we have C_{i} = C_{G_{i}} (1 ≤ i ≤ p) for some distinct G_{1}, …, G_{p} ∈ Λ̃. Using (7), we have (in X) U = X′ \ supp(M_{1} + ⋯ + M_{p}) where M_{i} ∈ Λ corresponds to G_{i} ∈ Λ̃ under the bijection Λ → Λ̃ defined in 1.6. Since X \ U has pure dimension one,

(8)  U = X \ supp(M_{1} + ⋯ + M_{p}).

Suppose that the reducible members F_{1}, …, F_{s} of Λ̃ have been labeled in such a way that

{F_{1}, …, F_{s}} \ {G_{1}, …, G_{p}} = {F_{1}, …, F_{r}} (where 0 ≤ r ≤ s).

Since U is connected at infinity, we may write (using (7) and 1.10):

(9)  X̃ \ U = supp(H + F_{1}^{u} + ⋯ + F_{r}^{u} + G_{1} + ⋯ + G_{p}).

Let γ : X̃ → S (where S is smooth) be the shrinking of G_{1}, …, G_{p} to 0-curves (see 1.2 and 1.9). Then U →^{γ} γ(U) is an isomorphism and S \ γ(U) = supp(D′), where D′ is a divisor of S with strong normal crossings. By (9), the dual graph G of (S, D′) is a tree with p+r branches at γ(H): p branches γ(G_{i}) consisting of a single vertex of weight zero and r branches γ(F_{i}^{u}) in which every weight is strictly less than −1. Thus G satisfies the hypothesis of Lemma 1.12 and part 1 of that result gives r = 0, so {F_{1}, …, F_{s}} ⊆ {G_{1}, …, G_{p}}. From this and (8), it follows that U satisfies condition (2). □

1.13. Let X be a complete normal rational surface.

Given an affine ruling Λ of X, consider (X̃, Λ̃) = (X, Λ)^{∼} and the divisor D of X̃ with strong normal crossings such that supp D = X̃ \ X′; let G(Λ) = G(D, X̃) (the dual graph of D in X̃).

Then the equivalence class of the weighted graph G(Λ) depends only on X and has a unique minimal element, say E_{X}. Indeed, let X̂ and Ê be as in the proof of Proposition 1.5, and let E_{X} be the dual graph of Ê in X̂; then the weighted graph E_{X} is the only minimal element of its equivalence class and G(Λ) contracts to E_{X}.

Definition 1.14. Let X be a complete normal rational surface and Λ an affine ruling of X. Define β(Λ) ∈ ℕ by:

β(Λ) = number of branch points of G(Λ) − number of branch points of E_{X},

where G(Λ) and E_{X} are as in 1.13. If β(Λ) = 0, we say that Λ is basic. Remark. In 1.13 and 1.14, if X satisfies (‡) (which includes the case where X is smooth), then E_{X} has no branch point and, consequently, Λ is basic if and only if the divisor D has no branching component. Theorem 1.15. Suppose that X satisfies (†). Then:

(1) at most one singular point of X is not a cyclic quotient singularity. Let Λ be an affine ruling of X and assume that at least one of the following conditions holds:

(i) X satisfies (‡); or (ii) β(Λ) > 0.

Let (X̃, Λ̃) = (X, Λ)^{∼}. Then the following hold:

(2) Λ̃ has at most two reducible members and one of them contains all branching components of X̃ \ X′.

(3) Sing(X) ∪ Bs(Λ) contains at most three points.

(4) Λ has at most two multiple members. Moreover, if {F_{1}, F_{2}} is a subset of Λ containing all multiple members (where F_{1} ≠ F_{2}), and if F_{i} = ν_{i}C_{i} (where C_{i} is a curve and ν_{i} ≥ 1, i = 1, 2), then:

(5) X \ (C_{1} ∪ C_{2}) is isomorphic to ℙ^{2} minus two lines.

(6) Pic(X_{s}) ≅ ℤ ⊕ ℤ/dℤ, where d = gcd(ν_{1}, ν_{2}).

Proof. Let X̂, Ê, Λ̂ and ρ : X̃ → X̂ be as in the proof of Proposition 1.5; let the notation be as in 1.10.

First, it is clear that the connected components of X̃ \ X′ are D_{0} and the nonempty F_{i}^{ℓ}; in particular, there are at most s + 1 such components and, taking images under X̃ → X̂ → X, we get that Sing(X) ∪ Bs(Λ) contains at most s + 1 points.

If Λ̂ has a base point, denote it by P ∈ X̂ and observe that ρ^{−1}(P) is connected and that H ⊆ ρ^{−1}(P) ⊆ supp(D); thus ρ^{−1}(P) ⊆ D_{0} and consequently the restriction of ρ to the open set X̃ \ D_{0} is an isomorphism. Of course, this is also the case if Λ̂ does not have a base point (ρ is the identity map). Since F_{i}^{ℓ} is contained in that open set, 1.10 implies:

ρ(F_{i}^{ℓ}) is either empty or an admissible chain (for each i = 1, …, s).

On the other hand, the connected components of Ê are among those of X̂ \ X′, and these are ρ(D_{0}) and the nonempty ρ(F_{i}^{ℓ}). So at most one connected component of Ê is not a ρ(F_{i}^{ℓ}); consequently, at most one connected component of Ê is not an admissible chain, i.e., (1) holds.

Recall that F_{i}^{u} meets H for all i = 1, …, s, so the branching number of H in D_{0} is precisely s. Assuming that (i) or (ii) holds, we will now show that s ≤ 2 and that assertion (2) of the Theorem holds. For this, we may assume that D (or equivalently D_{0}) has a branching component.

Note that ρ(D_{0}) is either a point or a connected component of Ê. Thus, under assumption (i), D_{0} contracts to an admissible chain or to a single point; since we assumed that D_{0} has a branching component, it follows that β(Λ) > 0. Hence, we may assume that (ii) holds.

Then D is not minimal, i.e., it has a component C which is not branching in D and which satisfies C^{2} = −1; since C ≠ H implies C^{2} < −1, we must have C = H, so H is not branching in D and s ≤ 2. In particular, supp(F_{1} + F_{2}) contains every branching component of D. If each of F_{1}, F_{2} contains a branching component of D then (since C^{2} < −1 for all components C of F_{1} +F_{2}) no contraction of D can decrease the number of branching components—contradicting the assumption that β(Λ) > 0. This proves assertion (2) of the Theorem. The other assertions easily follow from (2) and results 1.8 and 1.11. □

Corollary 1.16. If X satisfies (†) then at most one singular point of X is not a cyclic quotient singularity. If X satisfies (‡) then X has at most three singular points. 1.17. We prove the following statement, which was claimed without proof in the introduction: Given X satisfying (‡) and a curve C on X, the following are equivalent:

(i) κ̄(X_{s} \ C) = −∞; (ii) there exists at least one affine ruling Λ of X such that nC ∈ Λ for some n > 0. Condition (ii) clearly implies (i). If (i) is satisfied then we have to show that U =

X_{s} \ C is affine-ruled (then 1.11 implies (ii)). Consider X̃ → X̂ → X, where X̂ → X is the minimal resolution of singularities of X and X̃ → X is further blowing-up so that the inverse image C̃ of C has normal crossings. Then the complement of U in X̃ is a divisor D with normal crossings and every connected component of D other than C̃ is a linear chain. Since the divisor class group of X (≅ Pic X_{s}) has rank 1, any two curves on X meet. Hence, if E ⊂ X̃ is a curve meeting U which is shrunk in making (X̃, D) almost minimal, E meets C̃. Hence, on the almost minimal model, the boundary divisor again has at most one non-linear component. Since the connected component of the boundary containing C is not contractible, [15] implies that U is affine-ruled.

2. Modification of affine rulings

In the proof of the following theorem, we consider an arbitrary affine ruling Λ satisfying β(Λ) > 0 and “reduce” it to an affine ruling Λ′ such that β(Λ′) < β(Λ) (see Definition 1.14 for β). We will see later that this reduction process is an instance of a more general modification process. Theorem 2.1. If X satisfies (†) then it admits a basic affine ruling.

Proof. Suppose that Λ is an affine ruling of X satisfying β(Λ) > 0. Consider X′ = X_{s} \ Bs(Λ), (X̃, Λ̃) = (X, Λ)^{∼}, let D be the reduced effective divisor of X̃ such that X̃ \ X′ = supp(D), H the unique section of Λ̃ contained in D and X̃ →^{ρ} X̂ → X as in the proof of Proposition 1.5.

Since β(Λ) > 0, at least one component C of D satisfies:

(10)

C is branching in D and ρ(C) is either a point of X̂ or a curve not branching in ρ(D).

By Theorem 1.15, Λ̃ has at most two reducible members and one of them, say F̃, contains all branching components of D; in particular, C ⊂ supp(F̃). Recall from Proposition 1.8 that F̃ has a unique component CF̃ which meets X′. Consider the connected components Γ_{1} and Γ_{2} of supp(D + CF̃), where Γ_{1} contains H and Γ_{2} is either empty or an admissible chain of projective lines. Explicitely, if F̃ is the only reducible member of Λ̃ then Γ_{1} = supp(H + F̃) and Γ_{2} = ∅; if Λ̃ has two reducible members, say G_{1} = F̃ and G_{2}, then Γ_{1} = supp(H + G_{1} + G_{2}^{u}) and Γ_{2} = G_{2}^{ℓ} (see 1.10 for the notations G_{2}^{u} and G_{2}^{ℓ} and note that G_{2}^{ℓ} may be empty).

Consider the birational morphism m : X̃ → S which shrinks F̃ to a 0-curve (see 1.2, 1.9) and regard it as a composition X̃ = S_{n} ⟶^{m_{n}} ⋯ ⟶^{m_{1}} S_{0} = S of monoidal transformations. Since the exceptional locus of m has only one (−1)-component (namely, CF̃), the center of m_{i} is on the exceptional curve of m_{i−1} for each i > 1. It follows, in particular, that the unique component of F̃ which meets H is not branching in D, so C is not that component and m(C) is a point. Another consequence is that Γ_{1} has precisely three branches at C, say B, B^{u} and B^{ℓ}, where B contains CF̃, B^{u} contains H and every component of B^{ℓ} has self-intersection strictly less than −1.

Since m(C) is a point, we may factor m as X̃ → S̃ → S, where the image of C in S̃ is a (−1)-curve and is the first curve to be shrunk by S̃ → S. Then it is easy to see that X̃ → S̃ is the shrinking of B.

On the other hand, our choice of C (condition (10), above) allows us to factor ρ as X̃ ⟶^{α} U → X̂, where C̄ = α(C) is a curve, but is not branching in D̄ = α(D); then one sees that α is the shrinking of B^{u}. So we may consider a commutative diagram of smooth complete surfaces and birational morphisms:

(11)  \begin{matrix}X̃ & ⟶_{(B^{u})}^{α} & U & ⟵^{β} & Ω \\ ^{(B)}↓ & & ↓^{(B)} & ↓^{(B)} \\ S̃ & ⟶_{(B^{u})}^{γ} & S′_{+} & ⟵^{σ} & S′ \end{matrix}

where the labels (“B” or “B^{u}”) indicate what set is shrunk by each morphism—only the left square is being defined at this time. Let ν : X̃ → S′_{+} be the composition of these maps.

Let x be the self-intersection number of ν(C) in S′_{+}. Since the image of C in S̃ has self-intersection −1 and S̃ ⟶^{γ} S′_{+} increases that number by at least one, we have x ≥ 0. The dual graph of ν(Γ_{1}) in S′_{+} is:

(12)  [figure: …; x; ν(C); ω_{1}; ω_{q}; _{ν(B^{ℓ})}]

where q > 0, ω_{i} ≤ −2 and x ≥ 0.

Let P′_{+} ∈ S′_{+} be the unique point of ν(C) which also belongs to another component of ν(Γ_{1}) and consider the birational morphism S′ ⟶^{σ} S′_{+} obtained by blowing-up x times at P′_{+}, in such a way that the dual graph of σ^{−1}(ν(Γ_{1})) is:

(13)  [figure: …; …; 0; C′; −1; −2; −2; ω_{1}−1; ω_{2}; ω_{q}]

where the 0-curve C′ is the strict transform of ν(C). Since the morphism U → S′_{+} is isomorphic in a neighborhood of P′_{+}, the same sequence of x blowings-up can be

performed at the level of U; this defines a birational morphism Ω ⟶^{β} U and completes the definition of the above commutative diagram (11).

Note that each surface Y considered in this argument comes equipped with a birational transformation, say τ_{Y} : S′ → Y; consequently, the complete linear system |C′| on S′ (a ℙ^{1}-ruling of S′, by 1.2) determines a linear system (without fixed components) on each one of these surfaces. In particular, we will consider the linear systems Λ′ on X and Λ^{∗} on Ω defined in this way. Clearly, Λ^{∗} is a ℙ^{1}-ruling of Ω.

We claim that Λ′ is an affine ruling of X. For i = 1, 2, let Γ′_{i} = σ^{−1}(ν(Γ_{i})) ⊂ S′. Then the birational transformation τ_{X} : S′ → X restricts to an isomorphism μ′ going from the open subset W′ = S′ \ (Γ′_{1} ∪ Γ′_{2}) of S′ to the open subset X_{s} \ supp(F) of X (where F is the member of Λ which corresponds to F̃ under the bijection Λ → Λ̃ defined in (1.6)). Then Λ′ is the affine ruling of X determined by the X-immersion (S′, μ′) (see 2.3 for details).

We now argue that Λ and Λ′ have the same base point on X. Let D^{∗} = β^{−1}(D̄) and let C^{∗} be the strict transform of C̄ with respect to β. Note that α(H) is a point of C̄ and that the image of C̄ under U → X̂ → X is a point (because the image of D under X̃ → X̂ → X is a finite set of points); thus the base point of Λ is the image of C̄ under U → X̂ → X. On the other hand, consider the component H′ of Γ′_{1} which is a section of |C′| (if x > 0 (resp. x = 0) then H′ is the neighbor of the vertex of weight 0 in the graph (13) (resp. (12))); then the strict transform H^{∗} of H′ (with respect to Ω → S′) is a section of Λ^{∗} and satisfies H^{∗} ∩ C^{∗} ≠ ∅. Since H^{∗} and C^{∗} are components of D^{∗} and, under Ω → U → X̂ → X, D^{∗} is mapped to a finite set of points, we deduce that the image of H^{∗} in X coincides with that of C̄; so Λ and Λ′ have the same base point.

The morphisms X̃ ⟶^{α} U ⟵^{β} Ω give an isomorphism X̃ \supp(D) ≅ Ω\supp(D^{∗}); it follows that the birational morphism Ω ⟶^{β} U → X̂ → X restricts to an isomorphism from Ω \ supp(D^{∗}) to X_{s} \ Bs(Λ), which is equal to X_{s} \ Bs(Λ′) by the preceding paragraph. Since D^{∗} is a reduced effective divisor with s.n.c., (Ω, Λ^{∗}) ≻ (X, Λ′). Noting that the number of branching components of D^{∗} is strictly less than that of D, and taking into account assertion (4) of Proposition 1.5, we conclude that β(Λ′) < β(Λ). □

Formalization of the reduction process

Definition 2.2. Suppose that X is a complete normal rational surface. An Ximmersion is a pair (S, μ) where:

(1) S is a smooth complete surface and μ is an isomorphism from an open subset W of S to an open subset of X.

(2) S \ W is nonempty and is the support of a divisor (of S) with strong normal crossings.

(3) Exactly one of the connected components of S \W is a linear chain of projective lines with dual graph:

[figure: …; 0; x; ω_{1}; ω_{q}]

where q ≥ 0, ω_{i} ≤ −2 and x is any integer. We call this connected component the main component of (S, μ) and often denote it by Γ. We stress that Γ has at least two irreducible components, corresponding to the vertices of weights 0 and x in the above picture.

(4) If C is an irreducible component of S \ W which is not in the main component

Γ, then C^{2} ≤ −1 and if equality holds then C is branching in S \ W. By dom μ we mean the open set W; by the zero-component of (S, μ), we mean the component of Γ which corresponds to the pending vertex of weight 0 in (3).^{3} The neighbor of the zero-component (neighbor in the graph (3)) is called the section of (S, μ). If x = −1 in (3), we say that (S, μ) is in standard form. Remark. Let the assumptions and notations as in Definition 2.2. Then C ≅ ℙ^{1} for every irreducible component C of S \ W. This follows from 2.3, below: C ⊂ supp(Σ + Z_{1} + ⋯ + Z_{n}). 2.3. Let X be a complete normal rational surface. We claim that each X-immersion determines an affine ruling of X.

To see this, let (S, μ) be an X-immersion; let W = dom μ and let Γ, Z and Σ be the main component, zero-component and section of (S, μ) respectively. By 1.2, the complete linear system |Z| is a ℙ^{1}-ruling of S; also, Σ is a section of |Z|. Every irreducible component C of S \ W other than Σ satisfies C · Z = 0, so is contained in some member of |Z|. Consequently, we can choose a finite subset {Z_{1}, …, Z_{n}} of |Z| such that the open set

W_{0} = S \ supp(Σ + Z_{1} + ⋯ + Z_{n})

3Note that the pending vertex of weight 0 is not unique when q = 0 and x = 0; let us agree that an X-immersion always comes equipped with a choice of a zero-component.

is contained in W. Enlarging the set {Z_{1}, …, Z_{n}} if necessary, we may arrange that the morphism S → ℙ^{1} induced by |Z| restricts to a projection map W_{0} = Γ × 𝔸^{1} → Γ. It follows that, if we let Λ denote the linear system on X (without fixed components) determined by |Z| via μ, then Λ is an affine ruling of X.

We will describe the image of μ in the case where X satisfies (†); to do it, we need: Definition 2.4. Suppose that X satisfies (†), let Λ be an affine ruling of X and consider the pair (X̃, Λ̃) = (X, Λ)^{∼}. Let Λ̃_{∗} be the set of members F̃ of Λ̃ which satisfy:

(1) at most one member of Λ̃ \ {F̃} is reducible; and

(2) all branching components of X̃ \ X′ are in F̃.

We also define Λ_{∗} = {F ∈ Λ | F̃ ∈ Λ̃_{∗}}, where F ↦ F̃ is the bijection Λ → Λ̃ of Definition 1.6. 2.5. Note that, in Definition 2.4, Λ_{∗} ≠ ∅ if and only if Λ̃_{∗} ≠ ∅, if and only if Λ̃ has at most two reducible members and some member contains all branching components of X̃ \ X′. In particular, Theorem 1.15 implies:

(1) If X satisfies (‡) then Λ_{∗} is nonempty.

(2) If β(Λ) > 0, then Λ_{∗} has exactly one element. Lemma and definition 2.6. Suppose that X satisfies (†), let (S, μ) be an X-immer-sion and let Z and Γ be the zero-component and the main component of (S, μ) respectively.

(1) The complete linear system |Z| on S determines (via μ) an affine ruling Λ of X. Moreover, there is a unique F ∈ Λ_{∗} such that im μ = X_{s} \ supp(F). In this context, we say that (S, μ) determines (Λ, F).

(2) S \ dom(μ) has at most two connected components and, if it has two, the component other than Γ is an admissible chain.

Proof. In view of 2.3, the proof of assertion (1) will be complete if we can show that im μ = X_{s} \ supp(F) for some F ∈ Λ_{∗}.

Let W = dom μ and X′ = X_{s} \ Bs(Λ); since W is smooth and |Z| is base point free, μ(W) ⊆ X′. If μ(W) = X′ then (S, |Z|) ≻ (X, Λ) and, by part (4) of Proposition 1.5, there exists a birational morphism S → X̃ which restricts to an isomorphism W → X′. Let C ⊂ X̃ be the image of Z under S → X̃; then C is a component of D satisfying C^{2} ≥ 0, which is absurd. Hence μ(W) ⊂ X′ (strictly).

Consider (X̃, Λ̃) = (X, Λ)^{∼} and let the notations of 1.10 be in effect (in particular D, H and, given F ∈ Λ̃, C_{F}, F^{u} and F^{ℓ}). Regard μ(W) and X′ as open subsets of X̃. Observe that, in S, no connected component of S \ W can be shrunk to a smooth point; thus X̃ \ μ(W) has pure dimension one, so

X̃ \ μ(W) = supp(D + C_{1} + ⋯ + C_{p}) (p > 0)

where the C_{i} are distinct curves not contained in D.

In 2.3 we noted that the morphism S → ℙ^{1} determined by |Z| restricts to a projection map W_{0} = Γ × 𝔸^{1} → Γ, where W_{0} ⊆ W. Then, as in the proof of 1.11, 1.2 implies that C_{i} = C_{G_{i}} (1 ≤ i ≤ p) for some distinct G_{1}, …, G_{p} ∈ Λ̃. Let F_{1}, …, F_{r} (r ≥ 0) denote the reducible members of Λ̃ \ {G_{1}, …, G_{p}}. Define

(14)  G^{∘} = supp(H + G_{1} + ⋯ + G_{p} + F_{1}^{u} + ⋯ + F_{r}^{u}),
(15)  G = X̃ \ μ(W) = supp(G^{∘} + F_{1}^{ℓ} + ⋯ + F_{r}^{ℓ})

and note that G^{∘} and the nonempty F_{i}^{ℓ} are the connected components of G.

Let G be the dual graph of G and G^{∘} the dual graph of G^{∘} in X̃ (so G^{∘} is a connected component of G); let Q be the dual graph of S \ W in S and let Q^{∘} be the dual graph of Γ (so Q^{∘} is a connected component of Q). Clearly, Q and G are equivalent weighted graphs. Because no connected component of G or Q is equivalent to the empty graph, the connected components of Q correspond bijectively to those of G, in such a way that each component of Q is equivalent to the corresponding component of G. We claim that Q^{∘} corresponds to G^{∘} under that bijection. Indeed, Q^{∘} corresponds (and so is equivalent) to some connected component G′ of G; if G′ ≠ G^{∘} then G′ must be the dual graph of F_{i}^{ℓ} for some i, so every weight in G′ is strictly less than −1 and G′ is the unique minimal element of its equivalence class; consequently, Q^{∘} contracts to G′. This is absurd, because any contraction of Q^{∘} contains a nonnegative weight. So G^{∘} is equivalent to Q^{∘}, which is of the form (∗) described in Lemma 1.12.

By 1.2 (and 1.9), each G_{j} can be contracted to a 0-curve. Let \overline{G^{∘}} be the weighted graph obtained from G^{∘} by contracting all G_{j} to 0-curves; in view of (14), \overline{G^{∘}} has p + r branches at H: p branches consisting of a single vertex of weight zero and r branches in which every vertex has weight strictly less than −1. Thus part (2) of Lemma 1.12 implies that \overline{G^{∘}} is of the form (∗). So p = 1, r ≤ 1 and, if r = 1, H + F_{1}^{u} is a linear chain. So (15) simplifies to:

(16)  X̃ \ μ(W) = \begin{cases}supp(H + G_{1}), & if r = 0; \\ supp(H + G_{1} + F_{1}^{u} + F_{1}^{ℓ}), & if r = 1. \end{cases}

Since r is the number of reducible members of Λ̃\{G_{1}, …, G_{p}} = Λ̃\{G_{1}}, we have:

(17) At most one member of Λ̃ \ {G_{1}} is reducible.

Regarding (16), we observe: H is not branching in X̃ \μ(W); if F_{1}^{ℓ} is nonempty, then it is an admissible chain and a connected component of X̃ \μ(W); if r = 1 then H +F_{1}^{u} is a linear chain. Thus all branching components of X̃ \μ(W) are in G_{1} and in particular:

(18) All branching components of X̃ \ X′ are in G_{1}.

By (17) and (18), we obtain G_{1} ∈ Λ̃_{∗} and consequently M_{1} ∈ Λ_{∗}.

In X̃ we have μ(W) = X̃ \ supp(D + C_{1}) = X̃ \ supp(D + G_{1}) = X′ \ supp(G_{1}) so, in X, μ(W) = X′ \ supp(M_{1}) = X_{s} \ supp(M_{1}). This proves assertion (1). Assertion (2) follows from (16) and the argument concerning the connected components of G and Q. □

Definition 2.7. Let X be a complete normal rational surface.

(1) Let (S, μ) be an X-immersion, with zero-component Z and section Σ, and let W = dom μ. If P is a point of Z, we define an X-immersion (S′, μ′) = elm_{P} (S, μ) as follows: let π : S̃ → S be the blowing-up of S at P, Z̃ the strict transform of Z on S̃ and π′ : S̃ → S′ the contraction of Z̃. Let W′ = π′(π^{−1}(W)), consider the isomorphism θ : W′ → W obtained by restricting π ∘ (π′)^{−1} and define μ′ = μ ∘ θ. We say that (S′, μ′) is obtained from (S, μ) by an elementary transformation. We distinguish two types of elementary transformations: elm_{P} is of sprouting type (resp. of subdivisional type) if P ∈ Z \ Σ (resp. {P } = Z ∩ Σ). Note that, if (S′, μ′) = elm_{P} (S, μ), then (S, μ) = elm_{Q}(S′, μ′) for a suitable choice of a point Q; here, elm_{P} and elm_{Q} are of distinct types.

(2) Two X-immersions are equivalent if one can be obtained from the other by a sequence of elementary transformations.

(3) Given X-immersions (S, μ) and (S′, μ′), we write (S′, μ′) ≤ (S, μ) to indicate that (S′, μ′) is produced by performing on (S, μ) a sequence of elementary transformations of subdivisional type.

Proposition 2.8. Suppose that X satisfies (†).

(1) If Λ is an affine ruling of X and F ∈ Λ_{∗} then there exists an X-immersion (S, μ) which determines (Λ, F) (as in 2.6).

(2) Let (S, μ) and (S′, μ′) be X-immersions determining pairs (Λ, F) and (Λ′, F′) respectively. Then:

(Λ, F) = (Λ′, F′) if and only if (S, μ) is equivalent to (S′, μ′).

Proof. Let Λ be an affine ruling of X and F ∈ Λ_{∗}; let V = X_{s} \ supp(F). Consider (X̃, Λ̃) = (X, Λ)^{∼} and F̃ ∈ Λ̃_{∗} (recall the bijection Λ → Λ̃, F ↦ F̃, defined in 1.6). Since V ⊆ X′ = X_{s} \ Bs(Λ) and X′ can be viewed as a subset of X̃, we may write V = X′ \ supp(F̃). Let S be the surface obtained from X̃ by shrinking F̃ to a 0-curve (see 1.2, 1.9) and let m : X̃ → S be the corresponding morphism. Let W = m(V) and let μ : W → V be the restriction of m^{−1}. Then (S, μ) is an X-immersion and determines the pair (Λ, F); so (1) is proved.

The fact that equivalent X-immersions determine the same pair (Λ, F) is quite clear. Conversely, suppose that (S, μ) and (S′, μ′) are X-immersions determining the same pair (Λ, F); we will show that (S, μ) and (S′, μ′) are equivalent.

For (S, μ), we use the notations W, Z and Σ as in 2.7; for (S′, μ′), we use W′, Z′ and Σ′. Since μ and μ′ have the same image X_{s} \ supp(F), they determine a birational isomorphism S → S′ which restricts to an isomorphism W → W′. So there exists a smooth complete surface Ω and two birational morphisms, π : Ω → S and π′ : Ω → S′, such that if we regard π (resp. π′) as a composition of monoidal transformations then each one of these is centered at a point infinitely near S \ W (resp. S′ \ W′). We also assume that (Ω, π, π′) is minimal, i.e., that the total number of monoidal transformations in π and π′ is minimal. We denote this number by N((S, μ), (S′, μ′)). Since we assumed that (S, μ) and (S′, μ′) determine the same Λ, it follows that Σ̃ = Σ̃′, where Σ̃ (resp. Σ̃′) is the strict transform of Σ (resp. Σ′) on Ω.

If π is an isomorphism then S′ is obtained from S by contracting some irreducible components of S \ W; since no component of S \ W is a (−1)-curve except possibly Σ, and since π′ does not shrink Σ (for Σ̃ = Σ̃′), π′ must then be an isomorphism and we are done in this case.

Suppose that π is not an isomorphism; by the above paragraph (with π and π′ interchanged), π′ is not an isomorphism and we may consider a curve C̃ ⊂ Ω which is first to be shrunk by π′. By minimality of (Ω, π, π′), C̃ is the strict transform of some component C of S \ W satisfying C^{2} ≥ −1. Since π′ does not shrink Σ̃, we must have C = Z. Thus exactly one of the monoidal transformations making-up π has a center P which is a point of Z. It follows that

N(elm_{P} (S, μ), (S′, μ′)) < N((S, μ), (S′, μ′))

and we are done by induction. □

Definition 2.9. Suppose that X is a complete normal rational surface and that (S, μ) is an X-immersion. In this paragraph, we define a set Π(S, μ) of birational morphisms and, given π ∈ Π(S, μ), an X-immersion (S, μ) ∗ π determined by (S, μ) and π.

Let W = dom μ and let Γ, Z and Σ be the main component, zero-component and section of (S, μ) respectively.

Let Π(S, μ) be the set of birational morphisms π : S̃ → S, with S̃ smooth and complete, satisfying:

(1) the exceptional locus of π has a unique (−1)-component, which we denote E;

(2) π(E) is a point of Z \ Σ;

(3) π^{−1}(Γ) is a linear chain and E has two neighbors in it;

(4) one of the two branches of π^{−1}(Γ) at E can be shrunk to a smooth point (this must be the branch containing the strict transforms of Z and Σ; moreover, the first curve to shrink is either Z or Σ).

Given a point P of Z \ Σ, we also define

Π_{P} (S, μ) = {π ∈ Π(S, μ) | π is centered at P (i.e., π(E) = P)} .

Given π ∈ Π(S, μ), let γ : S̃ → S′_{+} be the birational morphism (with S′_{+} smooth) whose exceptional locus is the branch of π^{−1}(Γ) at E containing the strict transforms of Z and Σ. Note that γ is uniquely determined by π and that its exceptional locus has exactly one (−1)-component. Moreover, γ(E) is a curve whose self-intersection number is nonnegative; γ(Z) is a point of γ(E) and γ(E) is the only irreducible component of γ(π^{−1}(Γ)) containing that point; γ(π^{−1}(Γ)) is a linear chain with dual graph

[figure: …; x; γ(E); ω_{1}; ω_{q}] where ω_{i} ≤ −2, x ≥ 0 and q > 0. Consider the birational morphism σ : S′ → S′_{+} defined as follows:

(a) If x = 0, let S′ = S′_{+} and let σ be the identity map.

(b) If x > 0, let P′_{+} ∈ S′_{+} be the unique point of γ(E) which also belongs to another irreducible component of γ(π^{−1}(Γ)); define σ by blowing-up x times at P′_{+}, in such a way that the dual graph of σ^{−1}(γ(π^{−1}(Γ))) in S′ is:

[figure: …; …; 0; −1; −2; −2; ω_{1}−1; ω_{2}; ω_{q}]

where the 0-curve is the strict transform of γ(E). Then let W′ = σ^{−1}(γ(π^{−1}(W))) and let μ′ be the composite

W′ ⟶^{σ} γ(π^{−1}(W)) ⟶^{γ^{−1}} π^{−1}(W) ⟶^{π} W ⟶^{μ} μ(W).

Then (S′, μ′) is an X-immersion, determined by (S, μ) and π. We write (S′, μ′) = (S, μ) ∗ π and, informally, think of (S′, μ′) as the result of π “acting” on (S, μ). Note that μ and μ′ have the same image. Definition 2.10. Suppose that X satisfies (†).

(1) Let C be an equivalence class of X-immersions. Then C determines a pair (Λ, F) which, in turn, determines (X̃, Λ̃) = (X, Λ)^{∼} and F̃ ∈ Λ̃_{∗}. As shown in the first paragraph of the proof of Proposition 2.8, contracting F̃ to a 0-curve gives rise to an X-immersion (S, μ) which determines (Λ, F). We call (S, μ) the distinguished element of C.

(2) Suppose that Λ is an affine ruling of X such that β(Λ) > 0. Then (by 2.5) Λ_{∗} has exactly one element, say F, and we may consider the distinguished element (S, μ) of the equivalence class of X-immersions which determine (Λ, F). We call (S, μ) the standard X-immersion associated to Λ (it is an X-immersion in standard form). Note that (S, μ) comes equipped with a birational morphism m : X̃ → S (the contraction of F̃ to a 0-curve).

Corollary 2.11 (Reduction Theorem). Suppose that X satisfies (†) and that Λ is an affine ruling of X such that β(Λ) > 0. Consider the unique element F of Λ_{∗}, the standard X-immersion (S, μ) associated to Λ and the center P ∈ S of the birational morphism m : X̃ → S (the contraction of F̃ to a 0-curve). Then, for some π ∈ Π_{P} (S, μ), the pair (Λ′, F′) determined by the X-immersion (S, μ) ∗ π satisfies

β(Λ′) = β(Λ) − 1 and supp(F′) = supp(F).

Remark. In the conclusion of Corollary 2.11, we can replace “for some π ∈ Π_{P} (S, μ)” by “for every π ∈ Π_{P} (S, μ)”. This is because of part (4) of Lemma 4.4, which also implies that (Λ′, F′) is uniquely determined by Λ, i.e., is independent of the choice of π ∈ Π_{P} (S, μ).

Proof of 2.11. Let C be the branching component of D which is closest to H (notations D, H, etc as in the proof of 2.1); then it is easy to see that C satisfies the condition (10) of the proof. As in the proof of 2.1, factor m as X̃ → S̃ ⟶^{π} S and consider S ⟵^{π} S̃ ⟶^{γ} S′_{+} ⟵^{σ} S′. Then it is quite clear that π ∈ Π_{P} (S, μ) and that the Ximmersion (S′, μ′) constructed in the proof is exactly (S, μ)∗π. Then (S′, μ′) determines a pair (Λ′, F′) and the proof of 2.1 shows that β(Λ′) < β(Λ). Actually, we have β(Λ′) = β(Λ) − 1 because of how we chose C. □

Lemma 2.12. Let X be a complete normal rational surface. Suppose that I is an X-immersion, that π ∈ Π_{P} (I) and let J = I ∗π (where P is a point of the 0-component but not of the section of I). Let I^{−} (resp. J^{−}) denote the X-immersion obtained from I (resp. J) by performing one subdivisional elementary transformation.

(1) I^{−} ∗ π′ = J for some π′ ∈ Π_{P^{−}}(I^{−}), where P^{−} is the point, on the 0-component of I^{−}, which is the image of the strict transform of the 0-component of I.

(2) I ∗ π′ = J^{−}, for some π′ ∈ Π_{P} (I).

(3) If I′ ≤ I and J′ ≤ J then there exists π′ ∈ Π(I′) satisfying I′ ∗ π′ = J′.

(4) There exist I′ ≤ I, J′ ≤ J, π′ ∈ Π(I′) and π′′ ∈ Π(J′) satisfying

I′ ∗ π′ = J′ and J′ ∗ π′′ = I′.

Proof. Write I = (S, μ) and let Z and Σ be the 0-component and section of I; write J = (S′, μ′) = I ∗ π and consider S ←^{π} S̃ →^{γ} S′_{+} ←^{σ} S′, as in Definition 2.9.

To prove (1), consider the point {Q} = Z ∩ Σ, write (T, ν) = I^{−} = elm_{Q}(S, μ) and consider the commutative diagram

Ỹ ⟶ S̃ ⟶^{γ} S′_{+} ⟵^{σ} S′
↓^{π_{Y}}  ↓^{π}
T ⟵^{β} Y ⟶^{α} S

where α is the blowing-up at Q and β contracts the strict transform of Z. Then π′ = β ∘ π_{Y} ∈ Π_{P^{−}}(I^{−}) and I^{−} ∗ π′ = J (where P^{−} is the center of β).

To prove (2), let Z′ and Σ′ be the 0-component and section of J, consider the point {Q} = Z′ ∩ Σ′, write elm_{Q}(J) = J^{−} = (T, ν), let α : Y → S′ be the blowing-up of S′ at Q and β : Y → T the contraction of the strict transform of Z′. Consider the commutative diagram

Ũ ⟶^{ũ} U ⟶^{u} S̃ ⟶^{π} S
↓  ↓  ↓^{γ}
T ⟵^{β} Y ⟶^{α} S′ ⟶^{σ} S′_{+}

Then π′ = π ∘ u ∘ ũ ∈ Π_{P} (I) and I ∗ π′ = J^{−}.

Assertion (3) follows immediately from (1) and (2). To prove (4), consider the sections Σ and Σ′ of I and J respectively. In view of (3), we may assume that Σ^{2} < −1 and (Σ′)^{2} < −1. Then, in the diagram S ←^{π} S̃ →^{γ} S′_{+} ←^{σ} S′, the map σ is the identity map, γ ∈ Π(J) and J ∗ γ = I. □

Proposition 2.13. Let X be a complete normal rational surface and suppose that (S, μ) and (T, ν) are X-immersions. Then the condition im μ = im ν is equivalent to the existence of a sequence {(S_{j}^{∗}, μ_{j}^{∗})}_{j=0}^{n} of X-immersions satisfying:

(1) (S_{0}^{∗}, μ_{0}^{∗}) ≤ (S, μ) and (S_{n}^{∗}, μ_{n}^{∗}) ≤ (T, ν);

(2) for all j = 1, …, n, we have

(S_{j−1}^{∗}, μ_{j−1}^{∗}) ∗ π is equivalent to (S_{j}^{∗}, μ_{j}^{∗}), for some π ∈ Π(S_{j−1}^{∗}, μ_{j−1}^{∗}).

For the proof, we will need the following notations. Given X-immersions I_{j} = (S_{j}, μ_{j}) (j = 1, 2) such that im μ_{1} = im μ_{2}, let D(I_{1}, I_{2}) denote the set of triples (Ω, π_{1}, π_{2}) satisfying:

(1) Ω is a smooth complete surface and π_{1} : Ω → S_{1} and π_{2} : Ω → S_{2} are birational morphisms;

(2) π_{j} is centered at points of S_{j} \ dom μ_{j} (j = 1, 2) and π_{1}^{−1}(S_{1} \ dom μ_{1}) = π_{2}^{−1}(S_{2} \ dom μ_{2});

(3) the birational transformations π_{2}π_{1}^{−1} and μ_{2}^{−1}μ_{1}, from S_{1} to S_{2}, are equal. Note that D(I_{1}, I_{2}) is nonempty (because im μ_{1} = im μ_{2}) and that, given any (Ω, π_{1}, π_{2}) ∈ D(I_{1}, I_{2}), if one of π_{1}, π_{2} is an isomorphism then both π_{1}, π_{2} are.

Given a birational morphism f : U → V of smooth complete surfaces, let N(f) ≥ 0 be the number of monoidal transformations in f.

Given D = (Ω, π_{1}, π_{2}) ∈ D(I_{1}, I_{2}), let N(D) = N(π_{1}) + N(π_{2}). Also, let N(I_{1}, I_{2}) =

min_{D∈D(I_{1},I_{2})} N(D).

Proof of Proposition 2.13. Clearly, the existence of the sequence implies im μ = im ν.

For the converse, let I = (S, μ) and J = (T, ν) be X-immersions such that im μ = im ν and consider the set

I_{(I,J)} = {(I_{1}, I_{2}) | I_{1} ≤ (S, μ), I_{2} ≤ (T, ν), Σ_{1}^{2} < −1, Σ_{2}^{2} < −1} ,

where I_{j} is an X-immersion and Σ_{j} is its section. We proceed by induction on the natural number d(I, J) defined by

d(I, J) = min {N(I_{1}, I_{2}) | (I_{1}, I_{2}) ∈ I_{(I,J)}} .

If d(I, J) = 0 then I and J are equivalent; then it is easy to see that there exists an X-immersion (S_{0}^{∗}, μ_{0}^{∗}) satisfying both (S_{0}^{∗}, μ_{0}^{∗}) ≤ (S, μ) and (S_{0}^{∗}, μ_{0}^{∗}) ≤ (T, ν), so we are done in this case. From now-on, we assume that d(I, J) > 0.

Choose (I_{1}, I_{2}) ∈ I_{(I,J)} such that N(I_{1}, I_{2}) = d(I, J); write I_{j} = (S_{j}, μ_{j}), D_{j} = S_{j}\dom μ_{j} and let Z_{j} and Σ_{j} be the 0-component and section of I_{j}. Choose (Ω, π_{1}, π_{2}) ∈ D(I_{1}, I_{2}) such that N(Ω, π_{1}, π_{2}) = N(I_{1}, I_{2}).

S_{1} ⟵^{π_{1}} Ω ⟶^{π_{2}} S_{2}

We claim that (i) Neither of π_{1}, π_{2} is an isomorphism. (ii) For each j = 1, 2, the exceptional locus of π_{j} has a unique (−1)-component, say

E_{j} ⊂ Ω, and π_{j} is centered at a point of Z_{j}; also, π_{2}(E_{1}) = Z_{2} and π_{1}(E_{2}) = Z_{1}. Moreover, we claim that (I_{1}, I_{2}) and (Ω, π_{1}, π_{2}) can be chosen in such a way that the following conditions hold:

(iii) π_{j} is centered at a point of Z_{j} \ Σ_{j} (for j = 1, 2); (iv) E_{j} has two neighbors in π_{j}^{−1}(Γ_{j}) (for each j = 1, 2), where Γ_{j} ⊆ D_{j} is the main

component of (S_{j}, μ_{j}). If (i) is false then, as pointed out just before the proof, both π_{1}, π_{2} are isomorphisms;

this contradicts d(I, J) > 0, so (i) holds.

By (i), the exceptional locus of π_{1} has at least one (−1)-component; let E_{1} ⊂ Ω be such a component. Since N(Ω, π_{1}, π_{2}) = N(I_{1}, I_{2}), π_{2} does not contract E_{1}. So, π_{2}(E_{1}) is a non-branching component of D_{2} satisfying π_{2}(E_{1})^{2} ≥ −1 and consequently π_{2}(E_{1}) = Z_{2}. In particular, E_{1} is unique.

If the center of π_{1} is not on Z_{1} then the strict transform Z̃_{1} ⊂ Ω of Z_{1} satisfies Z̃_{1}^{2} = 0. Thus π_{2}(Z̃_{1}) is a component of D_{2} with nonnegative self-intersection number and, consequently, π_{2}(Z̃_{1}) = Z_{2}. This is impossible, because π_{2}(E_{1}) = Z_{2} and E_{1} ≠ Z̃_{1}. Thus the center of π_{1} is on Z_{1}. Then (ii) follows by symmetry in π_{1} and π_{2}.

For each j = 1, 2, let P_{j} ∈ Z_{j} be the center of π_{j}; define an X-immersion I′_{j} = (S′_{j}, μ′_{j}) ≤ I_{j} and a morphism π′_{j} : Ω → S′_{j} as follows:

• If P_{j} ∈ Z_{j} \ Σ_{j}, let I′_{j} = I_{j} and π′_{j} = π_{j};
• If P_{j} ∈ Z_{j} ∩ Σ_{j}, let I′_{j} = elm_{P_{j}}(I_{j}) and consider

Ω

↓^{π_{j}^{−}}
S_{j} ⟵^{α_{j}} S_{j}^{−} ⟶^{β_{j}} S′_{j}

where α_{j} is the blowing-up of S_{j} at P_{j}, β_{j} is the contraction of the strict transform of Z_{j} relative to α_{j} and π_{j}^{−} is defined by π_{j} = α_{j} ∘ π_{j}^{−}. Then set π′_{j} = β_{j} ∘ π_{j}^{−}.

Then (I′_{1}, I′_{2}) ∈ I_{(I,J)}, (Ω, π′_{1}, π′_{2}) ∈ D(I′_{1}, I′_{2}) and N(Ω, π′_{1}, π′_{2}) = N(Ω, π_{1}, π_{2}) = d(I, J). Moreover, the center of π′_{j} is a point of Z′_{j} \ Σ′_{j} (for each j = 1, 2), where Z′_{j} and Σ′_{j} are the 0-component and section of I′_{j} respectively. In other words, we may simply assume that (I_{1}, I_{2}) and (Ω, π_{1}, π_{2}) have been chosen in such a way that (iii) holds. Finally, (iv) follows immediately from (i–iii).

We proved that there exists (I_{1}, I_{2}) ∈ I_{(I,J)} and (Ω, π_{1}, π_{2}) ∈ D(I_{1}, I_{2}) satisfying N(Ω, π_{1}, π_{2}) = N(I_{1}, I_{2}) = d(I, J) and conditions (i–iv). We will now show that d(I_{1} ∗ π, I_{2}) < d(I, J) for some π ∈ Π(I_{1}), which will complete the proof.

If π_{1}^{−1}(Γ_{1}) is a linear chain then π_{1} ∈ Π(I_{1}) and I_{1} ∗ π_{1} = I_{2}, so we are done in this case.

Assume that π_{1}^{−1}(Γ_{1}) is not a linear chain and consider the branching component C of π_{1}^{−1}(Γ_{1}) which is closest to the strict transform Z̃_{1} ⊂ Ω of Z_{1}. Note that C is contained in the exceptional locus of π_{1}, for otherwise we would have C = Z̃_{1}, but Z̃_{1} is not branching in π_{1}^{−1}(Γ_{1}) (because Z_{1} has one neighbor in Γ_{1} and the center of π_{1} is one point). Also, π_{1}^{−1}(Γ_{1}) has exactly three branches at C, say B, B^{u} and B^{ℓ}, where B^{u} contains Z̃_{1} and B contains E_{1}. Note that Z̃_{1} and E_{1} are the only (−1)-components of π_{1}^{−1}(Γ_{1}) and that all other components have self-intersection strictly less than −1.

Since π_{2}(π_{1}^{−1}(Γ_{1})) is the linear chain Γ_{2}, we know that B^{u} can be shrunk to a point, i.e., we may factor π_{2} as Ω →^{α} U →^{β} S_{2}, where α is the contraction of B^{u}. We may also factor π_{1} as Ω →^{α_{1}} S̃_{1} →^{π} S_{1}, in such a way that α_{1}(C) is a (−1)-curve on S̃_{1}; then α_{1} is the contraction of B to a point and α_{1}(C) is the only (−1)-component of the exceptional locus of π. This gives the first of the following commutative diagrams of smooth complete surfaces and birational morphisms:

(19)  \begin{matrix}Ω & ⟶_{α}^{(B^{u})} & U & ⟶_{β} & S_{2} \\ ^{(B)}↓^{α_{1}} & ^{(B)}↓ \\ S̃_{1} & ⟶_{γ}^{(B^{u})} & S′_{+} \\ ↓ & π \\ S_{1} \end{matrix}  \begin{matrix}Ω′ & ⟶_{α′} & U & ⟶_{β} & S_{2} \\ ^{(B)}↓^{π′_{1}} & ^{(B)}↓ \\ S′ & ⟶_{σ} & S′_{+} \end{matrix}

where the labels (B) and (B^{u}) indicate which set is contracted by each morphism.

Note that π ∈ Π(I_{1}), so we may consider the X-immersion I′_{1} = (S′, μ′) = I_{1} ∗ π. Recall, from Definition 2.9, that the construction of I_{1}∗π involves a birational morphism σ : S′ → S′_{+} which is the composition of x monoidal transformations, where x ≥ 0 is the self-intersection number of the curve (γ ∘ α_{1})(C) ⊂ S′_{+}. Let α′ : Ω′ → U consist of the “same” x monoidal transformations as σ, but performed at the level of U. This gives the second diagram in (19).

Let π′_{2} = β ∘ α′ : Ω′ → S_{2}, then (Ω′, π′_{1}, π′_{2}) belongs to D(I′_{1}, I_{2}) but not necessarely to I_{(I′_{1},I_{2})}. Note that the section Σ′_{1} of I′_{1} satisfies (Σ′_{1})^{2} ≤ −1 and let I′′_{1} be the X-immersion obtained from I′_{1} by performing one subdivisional elementary transformation. Then (I′′_{1}, I_{2}) ∈ I_{(I′_{1},I_{2})}, so

(20)  d(I′_{1}, I_{2}) ≤ N(I′′_{1}, I_{2}) ≤ N(I′_{1}, I_{2}) + 2 ≤ N(Ω′, π′_{1}, π′_{2}) + 2.

We have d(I, J) = N(Ω, π_{1}, π_{2}) = |B| + N(π) + |B^{u}| + N(β) and N(Ω′, π′_{1}, π′_{2}) = N(π′_{1}) + N(α′) + N(β) = |B| + x + N(β), where |B| and |B^{u}| denote the numbers of irreducible components of B and B^{u}. So

d(I, J) − N(Ω′, π′_{1}, π′_{2}) = N(π) + |B^{u}| − x.

Note that the self-intersection numbers of α_{1}(C) ⊂ S̃_{1} and γ(α_{1}(C)) ⊂ S′_{+} are −1 and x respectively, so γ increases that number by x + 1. Since N(γ) = |B^{u}|, we must have x + 1 ≤ |B^{u}|, so

d(I, J) − N(Ω′, π′_{1}, π′_{2}) > N(π)

and, by (20),

d(I′_{1}, I_{2}) < d(I, J) − N(π) + 2.

It is easy to see that N(π) ≥ 3, so d(I′_{1}, I_{2}) < d(I, J) and we are done. □

Conclusion

2.14. Given a surface X satisfying (†), consider the directed graph 𝕃(X) whose vertices are the affine rulings of X and where, given vertices Λ and Λ′, we draw an arrow Λ → Λ′ if the following condition holds: There exists an X-immersion I and an element π of Π(I) such that (i) I determines (Λ, F) for some F ∈ Λ_{∗} and (ii) I ∗ π determines (Λ′, F′) for some F′ ∈ Λ′_{∗}.

Part (4) of Lemma 2.12 implies that if there is an arrow Λ → Λ′ then there is also an arrow Λ ← Λ′. Corollary 2.11 implies that each connected component of 𝕃(X) contains a basic affine ruling. Thus, if we want to describe all affine rulings of X, we have to solve the following two problems:

(1) Make a list of all basic rulings of X.

(2) Describe the set Π(S, μ), for each X-immersion (S, μ).^{4}

Each one of these problems is nontrivial. The first one is highly dependent on the surface X; [6] solves it for the weighted projective planes (so in particular for ℙ^{2}). The second problem turns out to be independent of the surface and is completely solved in sections 3 and 4 (see in particular Corollary 4.4). Remarks. Let X be a surface satisfying (†).

(1) One can show^{5} that an affine ruling Λ is an isolated vertex of 𝕃(X) if and only if Λ_{∗} = ∅. Thus, if we make the additional assumption that X satisfies (‡), then no vertex of 𝕃(X) is isolated (see 2.5).

(2) Let us temporarily agree that, given affine rulings Λ and Λ′ of X, the phrase “Λ and Λ′ have a common member” means that there exists a curve C ⊂ X and positive integers n and n′ satisfying nC ∈ Λ_{∗} and n′C′ ∈ Λ′_{∗}. Then Proposition 2.13 implies: Two affine rulings Λ and Λ′ of X are in the same connected component of 𝕃(X) if and only if there exists a sequence {Λ_{i}}_{i=0}^{n} of affine rulings of X such that, for each i < n, Λ_{i} and Λ_{i+1} have a common member.

4The point would be in particular to describe explicitely how to increase β(Λ). Section 5 includes a complete answer to this question, as the value of β is easily determined by inspecting the data contained in the “discrete part”.

5By part (1) of Proposition 2.8 and part (1) of Corollary 4.4.

3. Contraction of weighted trees

We assume familiarity with weighted graphs, their blowing-up and blowing-down. We stress that, in weighted graphs, we do not allow multiple edges between a given pair of vertices. The empty weighted graph is denoted Ø. A weighted tree without branch points is called a linear weighted tree or a linear chain. 3.1. Given weighted graphs G and G′, the symbol G ← G′ indicates that G′ is obtained from G by blowing-up once. In that case, if V (resp. V′) denotes the set of vertices of G (resp. G′) then V can be viewed as a subset of V′ and V′ \ V contains a single vertex, say e. We call e the vertex created by G ← G′; we also say that G is the blowing-down of G′ at e. If e has one neighbor v in G′, then v can be viewed as a vertex of G and G ← G′ is called the blowing-up of G at the vertex v. If e has two neighbors u and v in G′, then {u, v} is an edge of G and G ← G′ is called the blowing-up of G at the edge {u, v}.^{6} Also, if G is any weighted graph and G′ is the weighted graph obtained from G by adding an isolated vertex of weight −1, then we regard G′ as a blowing-up of G. 3.2. Two weighted graphs are equivalent if one can be obtained from the other by means of a finite sequence of blowings-up and blowings-down. We will use the symbol “ ∼ ” for equivalence of weighted graphs (and “ ≈ ” for equivalence of weighted pairs, Definition 3.9).

Blowing-up according to a tableau

3.3. Let G_{0} be a weighted graph, e_{0} a vertex of G_{0} and c ≥ p > 0 integers. By blowingup G_{0} at e_{0} according to \binom{p}{c}, we mean producing the sequence G_{0} ← ⋯ ← G_{n} defined as follows.

(1) Let G_{0} ← G_{1} be the blowing-up at e_{0} and let e_{1} be the vertex of G_{1} so created.

Define ( \begin{matrix}u_{1} & x_{1} \\ v_{1} & y_{1} \end{matrix} ) = ( \begin{matrix}e_{1} & p \\ e_{0} & c−p \end{matrix}).

(2) If i ≥ 1 is such that G_{i}, e_{i} and ( \begin{matrix}u_{i} & x_{i} \\ v_{i} & y_{i} \end{matrix}) have been defined, then:

(a) If y_{i} = 0 then we set n = i and stop.

(b) If y_{i} ≠ 0 then let G_{i+1} be the blowing-up of G_{i} at the edge {u_{i}, v_{i}}, let e_{i+1} be the vertex of G_{i+1} so created and define

(\begin{matrix}u_{i+1} & x_{i+1} \\ v_{i+1} & y_{i+1} \end{matrix}) = \begin{cases}(\begin{matrix}e_{i+1} & x_{i} \\ v_{i} & y_{i} − x_{i} \end{matrix}) & if x_{i} ≤ y_{i}, \\ (\begin{matrix}u_{i} & x_{i} − y_{i} \\ e_{i+1} & y_{i} \end{matrix}) & if x_{i} > y_{i}. \end{cases}

Remark. In 3.3 we have n ≥ 1, with equality if and only if p = c. Of the n blowings-up in G_{0} ← ⋯ ← G_{n}, only G_{0} ← G_{1} is a blowing-up at a vertex.

6In [7] and [11], a blowing-up at a vertex (resp. at an edge) is called “sprouting” (resp.

“subdivisional”).

Definition 3.4. Let G_{0} be a weighted graph, e_{0} a vertex of G_{0} and

T = (\begin{matrix}p_{1} & ⋯ & p_{k} \\ c_{1} & ⋯ & c_{k} \end{matrix})

a matrix such that p_{i} ≤ c_{i} are positive integers for all i.

We define the sequence G_{0} ← ⋯ ← G_{n} obtained by blowing-up G_{0} at e_{0} according to T by induction on k:

• If k = 0 (i.e., T is the empty matrix), then n = 0 (no blowing-up is performed).

• If k = 1, then G_{0} ← ⋯ ← G_{n} is defined in 3.3.

• If k > 1, then G_{0} ← ⋯ ← G_{n} is

G_{0} ← ⋯ ← G_{m−1} ← G_{m} ← G_{m+1} ← ⋯ ← G_{n},

where G_{0} ← ⋯ ← G_{m} is the sequence obtained by blowing-up G_{0} at e_{0} according to \binom{p_{1}}{c_{1}} and G_{m} ← ⋯ ← G_{n} is obtained by blowing-up G_{m} at e_{m} according to ( \begin{matrix}p_{2} & ··· & p_{k} \\ c_{2} & ··· & c_{k} \end{matrix}) (where e_{m} is the vertex of G_{m} created by G_{m−1} ← G_{m}).

Definition 3.5. A tableau is a matrix T = ( \begin{matrix}p_{1} & ··· & p_{k} \\ c_{1} & ··· & c_{k} \end{matrix}) whose entries are integers satisfying c_{i} ≥ p_{i} ≥ 1 and gcd(p_{i}, c_{i}) = 1 for all i = 1, …, k. We allow k = 0, in which case we say that T is the empty tableau and write T = 1. The set of all tableaux is denoted T . Given T ∈ T , let h(T) denote the number of columns of T which are different from \binom{1}{1}.

3.6. Let T′ = ( \begin{matrix}p′_{1} & ··· & p′_{k} \\ c′_{1} & ··· & c′_{k} \end{matrix} ) and T′′ = ( \begin{matrix}p′′_{1} & ··· & p′′_{k} \\ c′′_{1} & ··· & c′′_{k} \end{matrix} ) be two 2 × k matrices as in 3.4. We say

that T′ and T′′ are equivalent if there exists a k-tuple (r_{1}, …, r_{k}) of positive rational numbers satisfying \binom{p′_{i}}{c′_{i}} = r_{i}\binom{p′′_{i}}{c′′_{i}} for all i = 1, …, k. If this is the case then, given a weighted graph G_{0} and a vertex e_{0} of G_{0}, blowing-up G_{0} at e_{0} according to T′ or T′′ gives the same sequence G_{0} ← ⋯ ← G_{n}.

Clearly, each matrix T′ as above is equivalent to a unique tableau T ∈ T (see

3.5). Also, every Hamburger-Noether tableau HN = ( \begin{matrix}p_{1} & ··· & p_{k−1} & p_{k} \\ c_{1} & ··· & c_{k−1} & c_{k} \\ α_{1} & ··· & α_{k−1} & α_{k} \end{matrix} ) (as in the appen-

dix of [11]) determines a unique tableau \overline{HN} = ( \begin{matrix}\overline{p}_{1} & ··· & \overline{p}_{k−1} & \overline{p}_{k} \\ \overline{c}_{1} & ··· & \overline{c}_{k−1} & \overline{c}_{k} \end{matrix} ) ∈ T where (\overline{p}_{i}, \overline{c}_{i}) =

(p_{i}/ gcd(p_{i}, c_{i}), c_{i}/ gcd(p_{i}, c_{i})). 3.7. Consider an arbitrary sequence S : G_{0} ← ⋯ ← G_{n} of blowings-up of weighted graphs and, for i = 1, … n, let e_{i} be the vertex of G_{i} created by G_{i−1} ← G_{i}. Suppose that S satisfies the two conditions:

(1) if n ≥ 1 then G_{0} ← G_{1} is the blowing-up at a vertex e_{0}; and

(2) if n ≥ 2 then, for each i = 1, …, n − 1, G_{i} ← G_{i+1} is the blowing-up at the

vertex e_{i}, or at an edge incident to e_{i}. Then there exists a unique tableau T ∈ T such that S is the blowing-up of G_{0} at e_{0} according to T. Moreover, those two conditions are necessary for the existence of T.

Weighted pairs

Definition 3.8. If G is a nonempty weighted graph and v a vertex of G then we say that (G, v) is a weighted pair. Definition 3.9. Let (G, v) and (G′, v′) be weighted pairs.

Let us say, provisionally, that (G′, v′) is an elementary contraction of (G, v) if G′ is the blowing-down of G at some vertex e ≠ v and if the canonical inclusion V′ ↪ V maps v′ to v (where V and V′ are the sets of vertices of G and G′ respectively).

We say that (G, v) is equivalent to (G′, v′), written (G, v) ≈ (G′, v′), if there exists a sequence (G_{0}, v_{0}), …, (G_{n}, v_{n}) of weighted pairs satisfying (G_{0}, v_{0}) = (G, v), (G_{n}, v_{n}) = (G′, v′) and such that, for each i = 1, …, n, one of the following holds:

(1) (G_{i}, v_{i}) is an elementary contraction of (G_{i−1}, v_{i−1}); or

(2) (G_{i−1}, v_{i−1}) is an elementary contraction of (G_{i}, v_{i}). In the special case where condition (1) holds for all i = 1, …, n, we say that (G, v) contracts to (G′, v′) and we write (G, v) ≥ (G′, v′).

When (G, v) ≈ (G′, v′), we sometimes identify v with v′. Definition 3.10. A weighted pair (G, v) is called a linear pair if G is a linear weighted tree and v has at most one neighbor in G. Definition 3.11. A weighted pair (L, w) satisfies the condition (0) if L is a tree of the form

[figure: …; 0; −1; ω_{1}; ω_{m}] (m ≥ 0, ω_{i} ∈ ℤ, ω_{i} ≤ −2)

and if w is the vertex of weight 0. (Remark: Because w is uniquely determined by L, we will often use the symbol L to represent the pair (L, w). For instance, we will write (G, v) ≈ L, or we will say that the “weighted pair (G, v) is equivalent to a tree L satisfying the condition (0)”, when we mean that (G, v) ≈ (L, w)).

If L satisfies the condition (0), with notation as above, we define the transpose of L by

L^{t} : [figure: …; 0; −1; ω_{m}; ω_{1}]

We also define L^{t^{i}} (i ≥ 0) the obvious way: L^{t^{0}} = L and L^{t^{i+1}} = (L^{t^{i}})^{t}.

In the special case where either m = 0 or ω_{i} = −2 for all i, we say that L is degenerate.

We now state one of the main results of this paper. In condition (2) of the theorem, M(L) · \binom{1}{ν} is the product of the 2 × 2 matrix M(L) (defined in 3.21, below) with the column \binom{1}{ν}. For the proof, see Theorem 3.32.

Theorem 3.12. Let (G_{0}, e_{0}) be a weighted pair and \binom{p}{c} ∈ T , \binom{p}{c} ≠ \binom{1}{1}. Consider the blowing-up G_{0} ← ⋯ ← G_{n} of G_{0} at e_{0} according to \binom{p}{c} and let e_{n} be the vertex of G_{n} created by G_{n−1} ← G_{n}. Then the following are equivalent:

(1) (G_{n}, e_{n}) is equivalent to a linear pair; (2) (G_{0}, e_{0}) is equivalent to a tree L satisfying the condition (0) and \binom{p}{c} = M(L)·\binom{1}{ν}

for some integer ν ≥ 0. Moreover, suppose that these conditions are satisfied, let G_{n} ← ⋯ ← G_{n+ν} be the blowing-up of G_{n} at e_{n} according to the 2 × ν tableau ( \begin{matrix}1 & ··· & 1 \\ 1 & ··· & 1 \end{matrix} ) and let e_{n+ν} be the vertex created by G_{n+ν−1} ← G_{n+ν}. Then (G_{n+ν}, e_{n+ν}) is equivalent to L^{t}.

Preliminaries to the proof of Theorem 3.12

Notation 3.13 (Blowing-up as an action). Define a binary operation on the set T of

tableaux (see 3.5) by ( \begin{matrix}p_{1} & ··· & p_{k} \\ c_{1} & ··· & c_{k} \end{matrix} ) ( \begin{matrix}p_{k+1} & ··· & p_{ℓ} \\ c_{k+1} & ··· & c_{ℓ} \end{matrix} ) = ( \begin{matrix}p_{1} & ··· & p_{k} & p_{k+1} & ... & p_{ℓ} \\ c_{1} & ... & c_{k} & c_{k+1} & ··· & c_{ℓ} \end{matrix} ). Then T is actually

the free monoid on the set of columns \binom{p}{c} where p ≤ c are relatively prime positive integers.

Let (G_{0}, e_{0}) be a weighted pair and T ∈ T a tableau, consider the blowing-up G_{0} ← ⋯ ← G_{n} of G_{0} at e_{0} according to T and let e_{n} be the vertex of G_{n} created by G_{n−1} ← G_{n}. Then we will write (G_{0}, e_{0})T = (G_{n}, e_{n}). Hence, blowing-up is a right action of T on the set of weighted pairs. 3.14. Let (G, v) and (G′, v′) be weighted pairs and T ∈ T a tableau. If (G, v) ≈ (G′, v′), then (G, v)T ≈ (G′, v′)T. Hence, blowing-up is also a right action of T on the set of equivalence classes of weighted pairs.

3.15. Let G be a weighted graph, v_{1}, …, v_{n} its vertices and ω_{i} the weight of v_{i}. Recall that one defines the determinant of G by det(G) = det(−A), where A denotes the “intersection matrix” of G, i.e., the n × n matrix with entries A_{ii} = ω_{i} and, if i ≠ j, A_{ij} = 1 (resp. 0) if v_{i}, v_{j} are neighbors (resp. are not neighbors). Then det(G) is independent of the ordering of the vertices and, if G and G′ are equivalent weighted graphs, det(G) = det(G′). 3.16 ([11], A.14). Let G be a weighted tree, v a vertex of weight Ω(v) in G, G_{1}, …, G_{n} the branches of G at v and v_{i} the vertex of G_{i} which is a neighbor of v in G. If d_{i} = det G_{i} and d′_{i} = det(G_{i} − {v_{i}}), then

det G = −Ω(v) · d_{1} ⋯ d_{n} − ∑_{i=1}^{n} d_{1} ⋯ d_{i−1}d′_{i} d_{i+1} ⋯ d_{n} .

Notation 3.17. Let G be a linear weighted tree [figure: …; v_{1}; v_{n}] and v = v_{1}. Then the following abbreviation is very convenient:

det_{i}(G, v) = \begin{cases}det G, & if i = 0, \\ det(G − \{v_{1}, …, v_{i}\}), & if 0 < i < n, \\ 1, & if i = n, \\ 0, & if i > n. \end{cases}

3.18. Let the notation be as in 3.17 and let Ω(v_{j}) be the weight of v_{j}. Then, by 3.16,

det_{i}(G, v) = −Ω(v_{i+1}) det_{i+1}(G, v) − det_{i+2}(G, v) (0 ≤ i < n).

In particular, if Ω(v_{1}) = 0 then det_{2}(G, v) = − det G. 3.19. Recall that an admissible chain is a linear tree in which every weight is at most −2. Using 3.18, it is easy to see that every admissible chain has a strictly positive determinant; note, also, that Ø is the only admissible chain with determinant 1. We also recall the following fact, which follows easily from 3.16 and 3.18:

Let G be a linear weighted tree and e a vertex of G. Suppose that all weights in G are strictly negative, and that e is the only vertex of weight −1. Then:

• If e has two neighbors and both of them have weight −2, then det(G) ≤ 0.

• If det(G) > 0 then G contracts to an admissible chain.

• If det(G) = 1 then G contracts to Ø.

Notation 3.20 ([11], A.16). Given relatively prime positive integers a and b, define \binom{a}{b}^{∗} = \binom{x}{y}, where x and y are the unique nonnegative integers which satisfy

|\begin{matrix}x & a \\ y & b \end{matrix}| = −1 and x < a or y < b.

Definition 3.21. Given a weighted tree L satisfying the condition (0), we shall now define a 2 × 2 matrix M(L), and a subset T (L) of T . Let v denote the vertex of weight 0 in L and consider the relatively prime integers r_{0} > r_{1} ≥ 0 given by r_{0} = det_{2}(L, v) and r_{1} = det_{3}(L, v) (see 3.17 and 3.20 for notations). Then define

M(L) = (\begin{matrix}x & r_{0} − r_{1} \\ y & r_{0} \end{matrix}) , where (\begin{matrix}x \\ y \end{matrix}) = (\begin{matrix}r_{0} − r_{1} \\ r_{0} \end{matrix})^{∗} .

Note that L is completely determined by the second column of M(L).

If L is nondegenerate (resp. degenerate) then, for each integer ν ≥ 0 (resp. ν > 0), let T_{ν} temporarily denote the 2 × (ν + 1) matrix ( \begin{matrix}p & 1 & ... & 1 \\ c & 1 & ... & 1 \end{matrix} ), where \binom{p}{c} = M(L) · \binom{1}{ν}. Then T_{ν} ∈ T and the first column of T_{ν} is not \binom{1}{1}. Define

T (L) = {T_{ν} | ν ≥ 0 (resp. ν > 0)} .

Given k ∈ ℕ, we also define T_{k}(L) = \{T ∈ T | T \binom{1}{1}^{k} ∈ T (L)\} (so T_{0}(L) = T (L)).

Here, T \binom{1}{1}^{k} is a product in the monoid T .

In the following statement, we abbreviate det( [figure: ω_{1}; ω_{m}; …]) by det(ω_{1}, …, ω_{m}). Lemma 3.22. Let ω_{1}, …, ω_{m} ≤ −2 be integers (where m ≥ 1) and define

b = det(ω_{1}, …, ω_{m}),
a = det(ω_{2}, …, ω_{m}), a′ = det(ω_{1}, …, ω_{m−1}) (a = 1 = a′ if m = 1)
a′′ = det(ω_{2}, …, ω_{m−1}) (a′′ = 0 if m = 1).

Then:

(1) \binom{a}{b}^{∗} = \binom{a′′}{a′}, \binom{b−a}{b}^{∗} = \binom{b−a−a′+a′′}{b−a′} and \binom{b−a′}{b}^{∗} = \binom{b−a−a′+a′′}{b−a};

(2) det(ω_{1}, …, ω_{m−1}) = b−y and det(ω_{2}, …, ω_{m−1}) = a+x−y, where \binom{x}{y} = \binom{b−a}{b}^{∗}.

Proof. Lemma 3.6 of [7] gives aa′ − ba′′ = 1, 0 ≤ a′′ < min(a, a′) and max(a, a′) < b; this gives \binom{a}{b}^{∗} = \binom{a′′}{a′} and it also follows that | \begin{matrix}b−a−a′+a′′ & b−a \\ b−a′ & b \end{matrix} | = −1. Since b, b − a and b−a′ are positive integers, (b−a−a′+a′′)b = (b−a)(b−a′)−1 ≥ 0, so b−a−a′+a′′ ≥ 0 and we obtain the second equality of assertion (1). The third equality follows from the second by symmetry, i.e., by interchanging a and a′. Assertion (2) follows from (1). □

Lemma 3.23. Let c > p > 0 be relatively prime integers, let G be the weighted graph which consists of a single vertex v of weight zero, and let (G′, v′) = (G, v)\binom{p}{c}. Then G′ has two branches at v′, with determinants of subtrees as follows:

[figure: (G′, v′) :; …; …; v; v′; −1; _{p′′}; _{p}; _{c}; ^{p′}; _{c−p−p′+p′′}; ^{c−p′}; _{c}; _{c−p}]

where we define \binom{p′′}{p′} = \binom{p}{c}^{∗}.

Moreover, if we let (G′′, v′′) = (G′, v′)\binom{1}{N} (with N ≥ 1) then the connected component of G′′ \ {v′′} containing v and v′ is as follows:

[figure: …; …; v; v′; −1−N; _{Np(c−p)+1}; _{Ncp+1}; _{Nc^{2}}; ^{Nc(c−p)+1}]

Proof. This follows from 3.22 and from A.18.2 and A.18.3 of [11]. □

Lemma 3.24. If L is a tree satisfying the condition (0) then M(L^{t}) = M(L)^{t}.

Proof. Use the notation of 3.11 for L. If m ≥ 1, the result follows from Lemma 3.22; if m = 0, it is trivial. □

We recall two properties of weigthed graphs^{7} and state them in the language of weighted pairs. First, if a weighted graph is equivalent to a linear weighted graph, then it contracts to a linear weighted graph. For weighted pairs, one has: 3.25. If a weighted pair is equivalent to a linear pair, then it contracts to a linear pair.

For the second property, consider a sequence G_{0} ← ⋯ ← G_{n} of blowings-up of weighted graphs satisfying the two conditions of 3.7 and such that (G_{n}, e_{n}) contracts to a linear pair; then (G_{j}, e_{j}) contracts to a linear pair, for every j < n satisfying:

7The first of these two facts is proved in [1], I.4.13. We don’t know a reference for the second one.

G_{j} ← G_{j+1} is a blowing-up at a vertex. This can be conveniently expressed as part (1) of: 3.26. Let (G, v) be a weighted pair.

(1) If there exists T ∈ T such that (G, v)T contracts to a linear pair, then (G, v) contracts to a linear pair.

(2) (G, v) contracts to a linear pair if and only if (G, v)\binom{1}{1} contracts to a linear pair.

Definition 3.27. A weighted pair (L, w) satisfies the condition (+) if L is a tree of the form

[figure: …; α; ω_{1}; ω_{m}] (α > 0, m ≥ 0, ω_{i} ∈ ℤ, ω_{i} ≤ −2)

and if w is the vertex of positive weight. 3.28. If (G, v) ≈ (L, w) are weighted pairs and (L, w) satisfies the condition (+), then (G, v) ≥ (L, w).

The (straightforward) proof of 3.28 is left to the reader. Statement 3.29 follows immediately from 3.28: 3.29. Let C be an equivalence class of weighted pairs. Then:

(1) The class C contains a pair satisfying the condition (0) if and only if it contains one satisfying the condition (+).

(2) The class C contains at most one pair satisfying the condition (0) and at most

one pair satisfying the condition (+). Definition 3.30. A weighted pair (G, v) is contractible if it is equivalent to some pair (L, w) which satisfies the condition (0). Then L is unique (by 3.29) and we say that (G, v) is of type L. Lemma 3.31. If (G, e) is any weighted pair then at most one integer r ≥ 0 is such that (G, e)\binom{1}{1}^{r} is contractible.

Proof. It suffices to show that, if r > 0 and (L, w) satisfies (+), then (L′, w′) = (L, w)\binom{1}{1}^{r} does not contract to a pair which satisfies the condition (+). But this is trivial. □

Main result

Except for notation, the following is exactly the same as Theorem 3.12. Theorem 3.32. Let (G, e) be a weighted pair and \binom{p}{c} ∈ T , \binom{p}{c} ≠ \binom{1}{1}. Then the following are equivalent:

(1) (G, e)\binom{p}{c} contracts to some linear pair;

(2) (G, e) is contractible, and \binom{p}{c} = M(L) · \binom{1}{ν} for some integer ν ≥ 0, where L is

the type of (G, e). Moreover, if these conditions are satisfied then (G, e)\binom{p}{c}\binom{1}{1}^{ν} is equivalent to L^{t}.

Proof. If condition (1) holds then, by 3.26, (G, e) contracts to a linear pair (M, e) which has no vertex of weight −1, except possibly e:

[figure: (M, e) :; α; e; α_{1}; α_{k}; …; (k ≥ 0, α_{i} ≠ −1).]

We claim that (M, e) satisfies the condition (+). Indeed, let (M′, e′) = (M, e)\binom{p}{c}:

[figure: (M′, e′) :; α′; e; α_{1}; α_{k}; …; −1; e′; …; …]

Because p ≠ c, we know that M′ has two branches C and C′ at e′; let C be the one which contains e. Since (M′, e′) contracts to a linear pair (by 3.14 and 3.25), and since every vertex of C′ has weight at most −2, C must be equivalent to the empty graph. This implies that all α_{i} are negative, so α_{i} ≤ −2 for all i. Another consequence is that α′ = −1, because all vertices of C other than e have weight at most −2. We also have α′ ≤ α − 2, because \binom{p}{c} produces at least two blowings-up, the first blowing-up is at the vertex e and the second one is at an edge incident to e. We conclude that α > 0, so M satisfies the condition (+).

In view of 3.29, condition (1) implies that (G, e) is equivalent to a pair (L, e) satisfying the condition (0); thus, in order to prove that conditions (1) and (2) are equivalent, we may assume that (G, e) ≈ (L, e) :

[figure: (G, e) ≈ (L, e) =; …; 0; e; −1; ω_{1}; ω_{m}] (m ≥ 0 and ω_{i} ≤ −2).

Consider the integers r_{0} = det_{2}(L, e) and r_{1} = det_{3}(L, e) used in the definition of M(L). Write (L′, e′) = (L, e)\binom{p}{c}, then:

(21) [figure: (L′, e′) :; −q; u; −1; e′; e; −1; ω_{1}; ω_{2}; ω_{m}; …; …; …; _{c}; _{p}; _{r_{1}}; _{c}; _{r_{0}}]

where the numbers under the braces represent the determinants of the indicated subtrees of L′ (in particular, the p and c in the left part of the picture are obtained from 3.23). Note that the extra assumptions made for drawing this picture (e.g., m > 1) have no effect on the following argument.

Let B and B′ be the two branches of L′ at e′, where B is the one containing e; then, by 3.16, det B = cr_{0} − pr_{0} − cr_{1}. Now condition (1) of the Theorem is equivalent to det B = 1, hence to

(22)  |\begin{matrix}p & r_{0} − r_{1} \\ c & r_{0} \end{matrix}| = −1.

This holds if and only if \binom{p}{c} − ν\binom{r_{0}−r_{1}}{r_{0}} = \binom{r_{0}−r_{1}}{r_{0}}^{∗} for some ν ≥ 0, and this is equivalent to condition (2) of the Theorem. Hence, conditions (1) and (2) of the Theorem are equivalent.

Assume that conditions (1) and (2) hold; continuing with the same notation, there remains to prove that (L′, e′)\binom{1}{1}^{ν} is equivalent to L^{t}.

The pair (L′, e′) contracts to a linear pair (L′′, e′):

(23)  [figure: (L′′, e′) :; −q; u; α; e′; …; (α ≥ 0, q ≥ 2)]

where L′′ − {e′} is identical to the branch B′ of L′ at e′. Since B′ is nonempty and every weight in it is at most −2, we have

(24)  c = det_{1}(L′′, e′) > det_{2}(L′′, e′) > det_{3}(L′′, e′) ≥ 0,

where the equality comes from the fact that det B′ = c (see the picture at line (21)). By 3.18, det L = − det_{2}(L, e) = −r_{0}, so

(25)  det H = −r_{0},  for each weighted graph H equivalent to L.

So we have −r_{0} = det L′′ = −α det_{1}(L′′, e′) − det_{2}(L′′, e′), i.e.,

(26)  r_{0} = αc + det_{2}(L′′, e′).

We have to separate two cases.

\underline{Case α > 0}. Since det_{2}(L′′, e′) > 0, we have c < r_{0} by equation (26). From this and equation (22), we deduce that \binom{p}{c} = \binom{r_{0}−r_{1}}{r_{0}}^{∗} and hence that ν = 0. So we have to show that (L′′, e′) is equivalent to L^{t}.

Observe that (L′′, e′) ≈ (L^{(3)}, e′), where

[figure: (L^{(3)}, e′) :; −q−1; u; 0; e′; −2; −2; −1; …; …]

satisfies the condition (0) (L^{(3)} is obtained from L′′ by blowing-up α times). We have det_{2}(L^{(3)}, e′) = − det L^{(3)} by 3.18, and since det L^{(3)} = −r_{0} by equation (25),

(27)  det_{2}(L^{(3)}, e′) = r_{0}.

We have det_{1}(L^{(3)}, e′) = 1 · det_{2}(L^{(3)}, e′) − det_{3}(L^{(3)}, e′) = r_{0} − det_{3}(L^{(3)}, e′). Since the weighted trees L^{(3)} − {e′} and L′′ − {e′} are equivalent, we also have det_{1}(L^{(3)}, e′) = det_{1}(L′′, e′) = c. Thus

(28)  det_{3}(L^{(3)}, e′) = r_{0} − c.

From equations (27) and (28), we obtain that the second column of M(L^{(3)}) is \binom{c}{r_{0}}, which is identical to the second column of M(L)^{t} = M(L^{t}) (by 3.24). Hence, L^{(3)} = L^{t}, i.e., (G, e)\binom{p}{c}\binom{1}{1}^{ν} is equivalent to L^{t} in this case.

\underline{Case α = 0}. This time we have det_{2}(L′′, e′) = r_{0} by (26), and c = det_{1}(L′′, e′) = q det_{2}(L′′, e′) − det_{3}(L′′, e′) ; so, if we write ρ = det_{3}(L′′, e′),

c = qr_{0} − ρ (q ≥ 2, 0 ≤ ρ < r_{0}).

In particular we have c > r_{0}, so ν > 0. Since α = 0 and ν > 0, the pair (L^{(3)}, e′′) = (L′′, e′)\binom{1}{1}^{ν} looks like this:

(29)  [figure: (L^{(3)}, e′′) :; −q; u; −1; e′; …; …; −2; −2; −1; e′′]

and (L^{(3)}, e′′) ≥ (L^{(4)}, e′′) , where

[figure: (L^{(4)}, e′′) :; ν−q; u; 0; e′′; …]

On the other hand, if we write M(L) = ( \begin{matrix}x & r_{0}−r_{1} \\ y & r_{0} \end{matrix} ) then by definition of ν we have c = νr_{0} + y = (ν + 1)r_{0} − (r_{0} − y) with ν + 1 ≥ 2 and 0 ≤ r_{0} − y < r_{0}. So q = ν + 1 and ρ = r_{0} − y. In particular, (L^{(4)}, e′′) satisfies the condition (0).

Since L^{(4)} − {e′′, u} is identical to L′′ − {e′, u}, we have

det_{i}(L^{(4)}, e′′) = det_{i}(L′′, e′) (all i ≥ 2)

so, in particular,

det_{2}(L^{(4)}, e′′) = r_{0} and det_{3}(L^{(4)}, e′′) = ρ = r_{0} − y.

So the second column of M(L^{(4)}) is \binom{y}{r_{0}}, which is identical to the second column of M(L)^{t} = M(L^{t}). Hence, L^{(4)} = L^{t}, i.e., (G, e)\binom{p}{c}\binom{1}{1}^{ν} is equivalent to L^{t}. □

We now give some corollaries to Theorem 3.32. See Definition 3.21 for T (L) and T_{k}(L). Corollary 3.33. Let (G, e) be a weighted pair and T = \binom{p}{c}\binom{1}{1}^{r} ∈ T , where r ≥ 0 and \binom{p}{c} ≠ \binom{1}{1}. Then the following are equivalent:

(1) (G, e)T is contractible;

(2) (G, e) is contractible of type L and T ∈ T (L). Moreover, if these conditions hold then (G, e)T is equivalent to L^{t}. Remark. By definition of T (L) (3.21), r = 0 can occur if and only if L is nondegenerate.

Proof. Suppose that condition (1) holds. Then, in particular, (G, e)\binom{p}{c}\binom{1}{1}^{r} contracts to a linear pair, so (G, e)\binom{p}{c} contracts to a linear pair by 3.26. By Theorem 3.32, we obtain that (G, e) is equivalent to a tree L which satisfies the condition (0), that \binom{p}{c} = M(L) · \binom{1}{ν} for some ν ≥ 0 and that (G, e)\binom{p}{c}\binom{1}{1}^{ν} is equivalent to L^{t}. By Lemma 3.31 we get r = ν; hence, T ∈ T (L) and (G, e)T is equivalent to L^{t}.

The proof that (2) implies (1) is left to the reader. □

Notation 3.34. T^{#} = T \ \binom{1}{1}T

Hence, T^{#} contains the empty tableau, and all nonempty tableaux whose first column is not \binom{1}{1}. This is a submonoid of T with the property that each T ∈ T^{#} has a unique factorization into irreducibles: T = T_{r} ⋯ T_{1}, T_{i} ∈ T^{#}, h(T_{i}) = 1. Note also that T_{k}(L) ⊂ T^{#}, for any k ∈ ℕ and L satisfying the condition (0).

Iterating Corollary 3.33 gives:

Corollary 3.35. Let A be a weighted pair and T ∈ T^{#}. If T = T_{r} ⋯ T_{1} is the irreducible factorization of T in T^{#}, then the following are equivalent:

(1) AT is contractible of type L,

(2) A is contractible of type L^{t^{r}}, and T_{i} ∈ T (L^{t^{i}}) for all i = 1, …, r. Definition 3.36. A weighted pair P = (G, v) is pseudo-linear if v has exactly one neighbor v′ in G and the connected component Γ of G containing v has the form:

Γ : [figure: …; 0; v; x; v′; ω_{1}; ω_{n}] (n ≥ 0, x, ω_{i} ∈ ℤ, x ≤ −1, ω_{i} ≤ −2).

We also say that P is pseudo-linear of type (−1 − x, L), where L is the weighted pair (satisfying the condition (0)) obtained from the above picture by replacing the “x” by a “−1”. If P is pseudo-linear, with Γ as in the above picture, let P^{t} be the weighted pair obtained from P by changing the weights in Γ, so as to obtain

[figure: …; 0; v; x; v′; ω_{n}; ω_{1}],

and by leaving the other connected components unchanged. Note that if P is pseudolinear of type (k, L), then P^{t} is pseudo-linear of type (k, L^{t}).

If P is pseudo-linear of type (k, L) then any weighted pair equivalent to P is said to be pseudo-contractible, or pseudo-contractible of type (k, L) (note that the type is well-defined). If a weighted pair P is pseudo-contractible of type (k, L), then k ∈ ℕ; if k > 0, then P \binom{1}{1} is pseudo-contractible of type (k − 1, L).

As an immediate consequence of Corollary 3.35, we have: Corollary 3.37. Let P be a weighted pair and T ∈ T^{#} \ {1}. If T = T_{r} ⋯ T_{1} is the irreducible factorization of T in T^{#}, then the following are equivalent:

(1) P T is pseudo-contractible of type (k, L),

(2) P is pseudo-contractible of type (0, L^{t^{r}}) and

T_{i} ∈ \{ \begin{matrix}T_{k}(L^{t}), & if i = 1, \\ T (L^{t^{i}}), & for all i = 2, …, r. \end{matrix}

4. Description of the set Π(S, μ)

4.1. Let f : X → Y be a birational morphism of smooth complete surfaces and D a nonzero divisor of Y with strong normal crossings. We say that \overline{HN}(f, D) is defined if the following condition holds:

If center(f) ∩ supp(D) is nonempty then it is a single point P, P belongs to exactly one component Z of D and f^{−1}(P) contains exactly one (−1)-curve.

If this condition holds, then we define \overline{HN}(f, D) ∈ T as follows.

• If center(f) ∩ supp(D) = ∅, define \overline{HN}(f, D) = 1 (the empty tableau).

• If center(f) ∩ supp(D) = {P }, let E ⊂ X denote the unique (−1)-curve in f^{−1}(P) and choose local coordinates (ξ, η) of Y at P such that ξ is a local equation of Z. Consider the finite Hamburger-Noether tableau

HN = HN(E; ξ, η) = HN(f; ξ, η) = ( \begin{matrix}p_{1} & ⋯ & p_{k−1} & p_{k} \\ c_{1} & ⋯ & c_{k−1} & c_{k} \\ α_{1} & ⋯ & α_{k−1} & α_{k} \end{matrix} )

as defined in the appendix of [11]. Then HN uniquely determines a tableau \overline{HN} ∈ T (3.6) and \overline{HN} is independent of the choice of (ξ, η). We define \overline{HN}(f, D) = \overline{HN}. Note that \overline{HN}(f, D) = \overline{HN}(f, Z).

We state two important properties of \overline{HN}(f, D). Recall that G(D, Y) denotes the dual graph of D in Y.

(1) Consider the weighted pairs R = (G(D, Y), Z) and R′ = (G(f^{−1}(D), X), E), where we regard f^{−1}(D) as a reduced effective divisor (with strong normal crossings) of X, and where

E = \{ \begin{matrix}f^{−1}(Z), & if center(f) ∩ supp(D) = ∅, \\ the (−1)-curve in f^{−1}(P), & if center(f) ∩ supp(D) = \{P \}. \end{matrix}

Then R′ = R \overline{HN}(f, D).

(2) (a) Suppose that f factors as X ⟶^{α} S ⟶^{β} Y and that center(α) ∩ β^{−1}(D), if nonempty, belongs to a unique component of β^{−1}(D). Then

\overline{HN}(β ∘ α, D) = \overline{HN}(β, D) \overline{HN}(α, β^{−1}(D)).

(b) Conversely, given any factorization \overline{HN}(f, D) = BA with A, B ∈ T , there is an essentially unique way to factor f as X ⟶^{α} S ⟶^{β} Y such that center(α)∩β^{−1}(D), if nonempty, belongs to a unique component of β^{−1}(D), \overline{HN}(β, D) = B and \overline{HN}(α, β^{−1}(D)) = A.

Definition 4.2. Suppose that X is a complete normal rational surface and that I = (S, μ) is an X-immersion. Let Γ, Z and Σ be the main component, 0-component and section of I, respectively, and let D be the divisor of S, with strong normal crossings, with support S \ dom μ. We define two weighted pairs determined by I:

P(I) = (G(D, S), Z) and L(I) = (G(Γ, S), Z).

Note that L(I) is the connected component of P(I) containing the distinguished vertex; also, if Σ^{2} < 0 then P(I) and L(I) are pseudo-linear of type (−1−Σ^{2}, L) (3.36), where L is the weighted pair obtained from L(I) by replacing the weight of Σ by “−1”. 4.3. Suppose that X is a complete normal rational surface and that I = (S, μ) is an X-immersion. Let Γ, Z and Σ be the main component, 0-component and section of I, respectively.

Given any morphism π : S̃ → S satisfying conditions (1) and (2) of 2.9, the tableau \overline{HN}(π, Γ) (see 4.1) contains enough information to decide whether π also satisfies conditions (3) and (4) of 2.9. Indeed,

L(I) \overline{HN}(π, Γ) = (G(π^{−1}(Γ), S̃), E),

and we immediately see that conditions (3) and (4) are equivalent to (3′) \overline{HN}(π, Γ) = \binom{1}{1}^{r}\binom{p}{c}, for some r ≥ 0 and \binom{p}{c} ≠ \binom{1}{1}; (4′) L(I) \binom{1}{1}^{r}\binom{p}{c} contracts to some linear pair. Hence, Theorem 3.32 allows us to give a complete description of Π_{P} (S, μ) (see 2.9). In

particular, if condition (4′) holds then L(I) \binom{1}{1}^{r} is contractible. Note that this implies r = −1−Σ^{2}; it also follows that there exists an X-immersion (S_{0}, μ_{0}) in standard form, obtained from (S, μ) via a sequence of r elementary transformations of sprouting type, and there exists π_{0} ∈ Π(S_{0}, μ_{0}), such that (S, μ) ∗ π = (S_{0}, μ_{0}) ∗ π_{0}. Corollary 4.4. Suppose that X is a complete normal rational surface and that (S, μ) is an X-immersion; let Γ, Z and Σ be the main component, 0-component and section of (S, μ) respectively. Let P ∈ Z \ Σ.

(1) Π_{P} (S, μ) ≠ ∅ if and only if Σ^{2} < 0.

(2) If π ∈ Π(S, μ) then there exists an X-immersion (S_{0}, μ_{0}) ≥ (S, μ) in standard form satisfying (S, μ) ∗ π = (S_{0}, μ_{0}) ∗ π_{0} for some π_{0} ∈ Π(S_{0}, μ_{0}).

(3) Suppose that I = (S, μ) is in standard form. If L(I) is non-degenerate (resp. degenerate) then

Π_{P} (S, μ) = {π_{ν} | ν ∈ ℕ (resp. ν ∈ ℕ \ {0})} ,

where π_{ν} : S̃_{ν} → S is the unique birational morphism which is centered at P, whose exceptional locus has a unique (−1)-component, and which satisfies

\overline{HN}(π_{ν}, Γ) = M(L(I)) · \binom{1}{ν}.

Moreover, (i) the section Σ_{ν} of the X-immersion (S, μ) ∗ π_{ν} satisfies Σ_{ν}^{2} = −1−ν; and (ii) if I′ is an X-immersion equivalent to (S, μ)∗π_{ν} and in standard form then L(I′) is the transpose of L(I).

(4) Suppose that (S, μ) is in standard form. Given any π, π′ ∈ Π_{P} (S, μ), the Ximmersions (S, μ) ∗ π and (S, μ) ∗ π′ are equivalent.

Proof. Assertion (3) is a direct consequence of Theorem 3.32. Assertion (2) was pointed out in 4.3 and the “only if” part of (1) follows from (2). Observe that (3) implies, in particular, that Π_{P} (I) is nonempty whenever I is in standard form; the “if” part of (1) easily follows from this and part (1) of Lemma 2.12.

In view of (3), it suffices to prove (4) in the special case where π = π_{ν} and π′ = π_{ν+1}. Write J = I ∗π_{ν} and consider the X-immersion J^{−} obtained from J by performing one elementary transformation of subdivisional type. By part (2) of Lemma 2.12, there exists π′′ ∈ Π_{P} (I) such that I ∗ π′′ = J^{−}. By (3), the section of J has self-intersection −1 − ν, so that of J^{−} has self-intersection −1 − (ν + 1). Again by part (3), we have π′′ = π_{n} for some n and the section of J^{−} = I ∗ π′′ has self-intersection −1 − n. Hence, n = ν + 1. Consequently, I ∗ π_{ν+1} = I ∗ π′′ = J^{−} is equivalent to J = I ∗ π_{ν}. □

5. Description of affine rulings by discrete data

See 5.3, below, for an introduction to this section. 5.1. Let X be a surface satisfying (†) and Λ an affine ruling of X.

Let (X̃, Λ̃) = (X, Λ)^{∼} be as in Proposition 1.5 and recall that X′ = X_{s} \ Bs(Λ) ⊂ X is embedded in X̃ as the complement of a divisor D with strong normal crossings, and that exactly one component H of D is a section of Λ̃ (see 1.8). Let m = −H^{2} ≥ 1. In view of 1.2 and convention 1.9, there is a unique birational morphism π : X̃ → 𝔽_{m} which contracts each reducible member of Λ̃ to a 0-curve and whose exceptional locus is disjoint from H.

Assume that Λ_{∗} is nonempty. Each choice of an element F ∈ Λ_{∗} determines a factorization

X̃ ⟶^{π_{2}} S ⟶^{π_{1}} 𝔽_{m},

of π, where:

• π_{2} is the contraction of F̃ to a 0-curve, where F̃ ∈ Λ̃_{∗} is the image of F ∈ Λ_{∗} under the bijection Λ → Λ̃ of 1.6. (Note that π_{2} is the identity map when F̃ is irreducible, or equivalently when F is a reduced member of Λ.)

• If some member of Λ̃ \ {F̃} is reducible then it is unique (by definition of Λ_{∗}) and we denote it by G̃; if there is no such member, let G̃ be any member of Λ̃ \ {F̃}. Let π_{1} be the contraction of G̃ (or rather, of π_{2}(G̃)) to a 0-curve. (This gives π_{1} = id when every member of Λ̃ \ {F̃} is irreducible.)

We will sometimes refer to π_{1} and π_{2} as the pair of morphisms determined by (Λ, F).

Regard D_{2} = π_{2}(supp(F̃ + D)) as a reduced effective divisor of S (with strong normal crossings) and observe that it has no branching component (because F ∈ Λ_{∗}); note that Z_{2} = π_{2}(supp F̃) and Σ_{2} = π_{2}(H) are respectively a 0-component and a (−m)-component of D_{2}. The curve Σ_{1} = π_{1}(Σ_{2}) = π_{1}(π_{2}(H)) ⊂ 𝔽_{m} is the negative section of the standard ruling of 𝔽_{m}; also, Z_{1} = π_{1}(π_{2}(supp G̃)) and π_{1}(Z_{2}) are distinct members of that ruling and D_{1} = Z_{1} + Σ_{1} + π_{1}(Z_{2}) is a divisor of 𝔽_{m} with strong normal crossings.

For each i ∈ {1, 2}, the exceptional locus of π_{i} contains at most one (−1)-curve and, if π_{i} ≠ id, the center P_{i} of π_{i} is a single point and belongs to Z_{i} \ supp(D_{i} − Z_{i}). Thus we may consider T_{i} = \overline{HN}(π_{i}, D_{i}) ∈ T , as defined in 4.1. In this way, (Λ, F) determines a unique triple (m, T_{1}, T_{2}) ∈ ℤ^{+} × T × T , which we call the discrete part of (Λ, F) (or of (X, Λ, F)). Definition 5.2. (1) Given a triple (X, Λ, F), where X is a surface satisfying (†),

Λ is an affine ruling of X and F ∈ Λ_{∗}, the discrete part of (X, Λ, F) is the triple (m, T_{1}, T_{2}) defined in 5.1. The notation is disc(X, Λ, F) = (m, T_{1}, T_{2}). We sometimes call (m, T_{1}, T_{2}) the discrete part of (Λ, F).

(2) Given a surface X satisfying (†), 𝕋(X) denotes the set of disc(X, Λ, F) such that Λ is an affine ruling of X and F ∈ Λ_{∗}; 𝕋_{0}(X) ⊆ 𝕋(X) denotes the set of disc(X, Λ, F) such that Λ is a basic affine ruling of X and F ∈ Λ_{∗}.

5.3. Let X be a surface satisfying (†). Can a description of the set 𝕋(X) be regarded as a solution to Problem 1 for X? There are two difficulties:

(D1) X may admit affine rulings Λ such that Λ_{∗} = ∅, and 𝕋(X) contains no information about such rulings.

Note that if we assume that all basic affine rulings of X are known then, in particular, all Λ satisfying Λ_{∗} = ∅ are known (see 2.5); this is why (D1) did not cause problems in sections 2 and 4. In this section, however, (D1) can only be resolved by assuming that X satisfies (‡), in which case all Λ satisfy Λ_{∗} ≠ ∅ (by 2.5 again).

(D2) Given τ = (m, T_{1}, T_{2}) ∈ 𝕋(X), we need a method for constructing all (Λ, F) (on X) such that disc(X, Λ, F) = τ.

Paragraph 5.29, below, describes a method for constructing all (X′, Λ′, F′) such that disc(X′, Λ′, F′) = τ, and this is good enough for (D2) if one can prove that all such X′ are isomorphic to X. Thus Corollary 5.32 implies that, if X satisfies (‡), describing 𝕋(X) does solve Problem 1 for X.

Some of the results of this section (5.17, 5.22, 5.23, 5.39) describe 𝕋(X) in terms of 𝕋_{0}(X), or in terms of the subset min 𝕋(X) of 𝕋_{0}(X). So, given X satisfying (‡),

this section reduces Problem 1 to the problem of describing 𝕋_{0}(X) or min 𝕋(X).

Definition 5.4. (1) Let n ≥ 1. By a weighted n-tuple, we mean an ordered n-tuple S = (G, v_{1}, …, v_{n−1}) where G is a weighted graph and v_{1}, …, v_{n−1} are distinct vertices of G (when n = 1, S is a weighted graph; when n = 2, it is a weighted pair 3.8).

(2) Let S be a weighted n-tuple, with n ≥ 2. Given T ∈ T , we define a weighted ntuple ST and a weighted (n−1)-tuple S⊖T as follows. Write S = (G, v_{1}, …, v_{n−1}) and let (G′, e) denote the weighted pair (G, v_{1})T. Note that v_{2}, …, v_{n−1} can be regarded as vertices of G′ \ {e}. Then we define

ST = (G′, e, v_{2}, …, v_{n−1}) and S ⊖ T = (G′ \ {e}, v_{2}, …, v_{n−1}).

Remarks. Let S = (G, v_{1}, …, v_{n−1}) be a weighted n-tuple.

(1) When n = 2, the definition of ST given in 5.4 agrees with the one given in section 3.

(2) The above definition gives S1 = S and S ⊖ 1 = (G \ {v_{1}}, v_{2}, …, v_{n−1}) (where 1 is the empty tableau). So, given T, T′ ∈ T , S ⊖ T = ST ⊖ 1 and S ⊖ (T T′) = (ST) ⊖ T′.

(3) Let P and P′ be weighted pairs and T ∈ T . If P ≈ P′ then, by 3.14, P ⊖ T ∼ P′ ⊖ T (where “≈” (resp. “∼”) means equivalence of weighted pairs (resp. weighted graphs)).

Notation 5.5. Given x ∈ ℤ, let G_{(x)} denote the weighted triple (G, v_{1}, v_{2}), where G is

the weighted graph [figure: 0; v_{1}; x; 0; v_{2}] . 5.6. Consider the weighted pair S consisting of a single vertex of weight zero. For any T ∈ T , the condition

S ⊖ T has no branch point and every weight in it is strictly less than −1 holds if and only if one of the following holds:

(1) T = 1;

(2) T = \binom{p}{c}, where \binom{p}{c} ≠ \binom{1}{1};

(3) T = ( \begin{matrix}p & 1 \\ c & N \end{matrix} ), where \binom{p}{c} ≠ \binom{1}{1} and N ≥ 1. 5.7. Let x be a negative integer and T_{1}, T_{2} ∈ T .

(1) The condition

G_{(x)} ⊖ T_{1} is pseudo-linear

holds if and only if T_{1} satisfies one of conditions (1–3) of 5.6. Moreover, if G_{(x)} ⊖T_{1} is pseudo-linear then it has at most two connected components and the one which does not contain the distinguished vertex is an admissible chain.

(2) The condition

(G_{(x)} ⊖ T_{1}) ⊖ T_{2} has no branch point and every weight in it, except possibly that of the middle vertex of G_{(x)}, is strictly less than −1,

holds if and only if each of T_{1}, T_{2} satisfies one of conditions (1–3) of 5.6.

Proof. To prove (1), write G_{(x)} = (G, v_{1}, v_{2}) and consider the weighted pair S = ({v_{1}}, v_{1}) (a single vertex of weight 0). We may regard S ⊖ T as the graph obtained from the weighted pair P = G_{(x)} ⊖ T by deleting the distinguished vertex (i.e., v_{2}), its unique neighbor and all edges incident to these two vertices. Note that P has at most two connected components, say L and A, where L contains the distinguished vertex and A is a (possibly empty) admissible chain. If P is pseudo-linear, S ⊖ T has no branch point (otherwise L would have one) and every weight in S ⊖ T is strictly less than −1; thus (by 5.6) T satisfies one of conditions (1–3) of 5.6. The converse is equally trivial, as is assertion (2). □

Notation 5.8. Given T ∈ T satisfying one of the conditions (1–3) of 5.6, we define Ť ∈ T as follows:

Ť = \{ \begin{matrix}1, & if T satisfies 5.6.1; \\ \binom{p′}{c}, & if T satisfies 5.6.2, where p′ is given by \binom{p′′}{p′} = \binom{p}{c}^{∗} (see 3.20); \\ ( \begin{matrix}c−p & 1 \\ c & N \end{matrix} ) , & if T satisfies 5.6.3. \end{matrix}

Note that if T satisfies condition 5.6.i (where i ∈ {1, 2, 3}) then so does Ť. If s is a positive integer, write T^{(ˇ s)} = (T^{(ˇ (s−1))})ˇ , where T^{(ˇ 0)} = T. Note that T^{(ˇ 2)} = T.

Lemma 5.9. Let (m, T_{1}, T_{2}) be the discrete part of (X, Λ, F), where X is a surface satisfying (†), Λ is an affine ruling of X and F ∈ Λ_{∗}.

(1) The weighted pair P = G_{(−m)} ⊖ T_{1} is isomorphic to P(I) (see 4.2), where I is the distinguished element of the equivalence class of X-immersions determining (Λ, F). In particular, P is pseudo-linear of type (m − 1, L) for some L; moreover, P has at most one connected component A other than the one containing the distinguished vertex, and A is an admissible chain.

(2) There is an isomorphism of weighted graphs (G_{(−m)} ⊖ T_{1}) ⊖ T_{2} → G(Λ) which maps the middle vertex of G_{(−m)} to the vertex H of G(Λ) (see 1.13 for the definition of G(Λ); H denotes the unique component of X̃ \X′ which is a section of Λ̃).

Proof. Let the notation (S, D_{2}, etc.) be as in 5.1. By definition of I (2.10), we have I = (S, μ) for some μ and, moreover, S \ dom μ = supp(D_{2}). So we have P(I) = (G(D_{2}, S), Z_{2}). For each i = 1, 2, let

E_{i} = \{ \begin{matrix}the unique (−1)-curve in π_{i}^{−1}(P_{i}), & if π_{i} ≠ id, \\ Z_{i}, & if π_{i} = id. \end{matrix}

Consider the weighted triple (G(D_{1}, 𝔽_{m}), Z_{1}, π_{1}(Z_{2})) = G_{(−m)}. Since π_{1}^{−1}(supp(D_{1})) = supp(D_{2}) ∪ E_{1} and E_{1} is not a component of D_{2}, we have G_{(−m)} ⊖ T_{1} = P(I) and (1) holds. Since π_{2}^{−1}(D_{2}) = supp(D) ∪ E_{2} and E_{2} is not a component of D, P(I) ⊖ T_{2} = G(D, X̃) = G(Λ). □

Notation 5.10. (1) Let 𝕋 be the set of triples (m, T_{1}, T_{2}) ∈ ℤ^{+} × T × T such that

T_{2} ∈ T^{#} (3.34) and T_{1} satisfies one of the conditions (1–3) of 5.6.

(2) Let 𝕋(†) be the set of (m, T_{1}, T_{2}) ∈ 𝕋 such that the intersection matrix (3.15) of the weighted graph (G_{(−m)} ⊖ T_{1}) ⊖ T_{2} is negative definite.

The following says, in particular, that 𝕋(X) ⊆ 𝕋(†) for each X satisfying (†). Lemma 5.11. Let (m, T_{1}, T_{2}) be the discrete part of (X, Λ, F), where X is a surface satisfying (†), Λ is an affine ruling of X and F ∈ Λ_{∗}. Then (m, T_{1}, T_{2}) ∈ 𝕋(†) and the following are equivalent:

(1) X satisfies (‡) and Λ is basic;

(2) T_{2} satisfies one of the conditions (1–3) of 5.6.

Proof. By 5.7 and part (1) of 5.9, T_{1} satisfies one of the conditions (1–3) of 5.6.

By part (2) of Lemma 5.9, every vertex of (G_{(−m)} ⊖ T_{1}) ⊖ T_{2}, except possibly the middle vertex of G_{(−m)}, has weight strictly less than −1. Write G_{(−m)} = (G, v_{1}, v_{2}) and note that the distinguished vertex v_{2} of the weighted pair G_{(−m)} ⊖ T_{1} has weight 0. If T_{2} ∉ T^{#} then T_{2} ≠ 1 and the first column of T_{2} is \binom{1}{1}, so the weight of v_{2} in (G_{(−m)} ⊖ T_{1}) ⊖ T_{2} is −1, contradicting the above observation. Hence, T_{2} ∈ T^{#}.

Let X̂ → X be the minimal resolution of singularities of X and let Ê ⊂ X̂ be the exceptional locus; since X is normal, the divisor Ê has a negative definite intersection matrix; since (G_{(−m)} ⊖ T_{1}) ⊖ T_{2} ≅ G(Λ) by 5.9, and G(Λ) contracts to G(Ê, X̂), we get (m, T_{1}, T_{2}) ∈ 𝕋(†).

By 5.7, (G_{(−m)} ⊖ T_{1}) ⊖ T_{2} (hence G(Λ)) has no branch point if and only if T_{2} satisfies one of the conditions (1–3) of 5.6. Hence, (1) and (2) are equivalent. □

Definition 5.12. Given (n, T_{1}, T_{2}), (m, T′_{1}, T′_{2}) ∈ 𝕋, write (n, T_{1}, T_{2}) ≡ (m, T′_{1}, T′_{2}) to indicate that

(G_{(−n)} ⊖ T_{1})T_{2} ≈ (G_{(−m)} ⊖ T′_{1})T′_{2}

(equivalence of weighted pairs). Note that “≡” is an equivalence relation on the set 𝕋. Theorem 5.13. Let τ, τ′ ∈ 𝕋 be such that τ ≡ τ′. Suppose that τ = disc(X, Λ, F), where X is a surface satisfying (†), Λ is an affine ruling of X and F ∈ Λ_{∗}. Then there exist an affine ruling Λ′ of X and F′ ∈ Λ′_{∗} such that τ′ = disc(X, Λ′, F′) and supp(F′) = supp(F).

In view of 2.14, the above result relates the viewpoint of this section with the operation “∗” of sections 2 and 4. See also Proposition 5.23.

The proof requires 5.14 and 5.15: Lemma 5.14. If P is a pseudo-linear weighted pair then:

(1) At most one pair (x, T) ∈ ℤ × T satisfies G_{(x)} ⊖ T = P.

(2) Suppose that G_{(x)} ⊖ T = P. Then T satisfies one of conditions (1–3) of 5.6 and G_{(x)} ⊖ Ť = (G_{(x)} ⊖ T)^{t} = P^{t}.

Proof. Write P = (G, v). We may assume that P = G_{(x)} ⊖ T for some (x, T). Then v has a unique neighbor v′ in G, and the weight of v′ is x; hence, x is uniquely determined. By 5.7, T satisfies one of conditions (1–3) of 5.6 (which proves part of assertion (2)). Note also that G has either one or two connected components; we say that G has two connected components, L and A, where L contains v and A is a (possibly empty) admissible chain. Moreover, L is as follows:

L : [figure: …; 0; v; x; v′; ω_{1}; y_{1}; ω_{n}; y_{n}] (n ≥ 0, ω_{i} ∈ ℤ, ω_{i} ≤ −2).

We now show that T is unique. If n = 0 (resp. n = 1) then T must be 1 (resp. \binom{1}{−ω_{1}}), so we may assume that n ≥ 2. We consider two cases.

If A is nonempty and contains a weight other than −2, then T = \binom{p}{c} for some p, c satisfying 1 < p < c. Then Lemma 3.23 implies that c = det(ω_{1}, …, ω_{n}) and p = det(ω_{2}, …, ω_{n}), so T is unique (notation as in Lemma 3.22).

Before treating the second case, let us observe that at most one i ∈ {1, …, n} can satisfy det(ω_{1}, …, ω_{i−1}) = det(ω_{i+1}, …, ω_{n}), because the left-hand-side is a strictly increasing function of i, while the right-hand-side is strictly decreasing.

If A is a chain of N−1 vertices of weight −2 (where N ≥ 1), then T = ( \begin{matrix}p & 1 \\ c & N \end{matrix} ), for some \binom{p}{c} ≠ \binom{1}{1}. Consider the vertex e (of weight −1) which is deleted from G_{(x)}T in order to define G_{(x)}⊖T; then e has a unique neighbor among {y_{1}, …, y_{n}}, say y_{j}. By Lemma 3.23 applied to the first column \binom{p}{c} of T, we have det(ω_{1}, …, ω_{j−1}) = c = det(ω_{j+1}, …, ω_{n}) and det(ω_{2}, …, ω_{j−1}) = p. So j must be the unique i of the preceding paragraph; since j is uniquely determined, so are c = det(ω_{1}, …, ω_{j−1}) and p = det(ω_{2}, …, ω_{j−1}). This proves assertion (1).

Assertion (2) is obtained from the following observation, which is a consequence of Lemma 3.23: Let (G, v) be the weighted pair consisting of a single vertex of weight 0, let \binom{p}{c} ∈ T , \binom{p}{c} ≠ \binom{1}{1}, and consider the weighted pair (G′, v′) = (G, v)\binom{p}{c}. Use the following notation for the weights in (G′, v′):

[figure: (G′, v′) = (G, v)\binom{p}{c} :; …; …; ω_{1}; ω_{n}; −1; a_{1}; a_{m}; v; v′].

If we define p′ by \binom{p′′}{p′} = \binom{p}{c}^{∗}, then:

[figure: (G, v)\binom{p′}{c} :; …; …; ω_{n}; ω_{1}; −1; a_{m}; a_{1}; v] .

On the other hand,

[figure: (G, v)\binom{c−p}{c} :; …; …; a_{m}; a_{1}; −1; ω_{n}; ω_{1}; v] .

□

5.15. Let S be a smooth complete surface, D a divisor of S with strong normal crossings and such that each component of D is rational, and G = G(D, S), the dual graph of D in S. Let also G′ be a weighted graph.

(1) Suppose that G can be contracted to G′. Let v_{1}, …, v_{n} be the vertices of G which disappear in that process and let D_{1}, …, D_{n} be the corresponding components of D. Then there is an essentially unique birational morphism π : S → S′ whose exceptional locus is D_{1}∪⋯∪D_{n} (where S′ is a smooth complete surface). Then the divisor D′ = π(D) of S′ with strong normal crossings has dual graph G′.

(2) Suppose that G′ can be contracted to G. Then there exists a (not necessarely unique) birational morphism π : S′ → S (where S′ is a smooth complete surface) such that the divisor D′ = π^{−1}(D) of S′, with strong normal crossings, satisfies G(D′, S′) = G′. The exceptional locus of π consists of the components of D′ corresponding to the vertices of G′ which disappear in the contraction to G.

In case (1) (resp. (2)), we call π simply “the (resp. a) birational morphism corresponding to G ≥ G′ (resp. G′ ≥ G)”; it is tacitely assumed that the above conditions are satisfied. Similar remarks hold if both G and G′ are weighted pairs; in this case, we have the additional information that π does not shrink the curve which corresponds to the distinguished vertex.

Proof of Theorem 5.13. Write τ = (n, T_{1}, T_{2}) and τ′ = (m, T′_{1}, T′_{2}).

Consider (X̃, Λ̃) = (X, Λ)^{∼} and recall that X \ (Sing X ∪ Bs Λ) is embedded in X̃ as the complement of a divisor D with strong normal crossings. Also, consider the curve C = CF̃ in X̃ (notation as in 1.8), where F̃ ∈ Λ̃ corresponds to F ∈ Λ via the bijection Λ → Λ̃ (1.6). Then we have (G_{(−n)} ⊖ T_{1})T_{2} = (G(D + C, X̃), C).

Since τ ≡ τ′, we have (G(D + C, X̃), C) ≈ (G_{(−m)} ⊖ T′_{1})T′_{2}; this can be written as (G(D + C, X̃), C) ≤ P ≥ (G_{(−m)} ⊖ T′_{1})T′_{2}, where P is a weighted pair and the inequalities indicate contractions of weighted pairs. In view of 5.15 we may consider a diagram X̃ ←^{ω} Ω →^{ω′} Ỹ, where Ω and Ỹ are smooth complete surfaces and ω and ω′ are birational morphisms corresponding to (G(D + C, X̃), C) ≤ P and P ≥ (G_{(−m)} ⊖ T′_{1})T′_{2} respectively. Define D′ = ω′(ω^{−1}D) and C′ = ω′(C̃), where C̃ ⊂ Ω is the strict transform of C. Then D′ + C′ is a divisor of Ỹ with strong normal crossings and (G_{(−m)} ⊖ T′_{1})T′_{2} = (G(D′ + C′, Ỹ), C′). Moreover, X̃ ← Ω → Ỹ gives an isomorphism Ỹ \ supp D′ → X̃ \ supp D which maps C′ onto C.

Since (G(D′+C′, Ỹ), C′) = (G_{(−m)}⊖T′_{1})T′_{2}, the weighted graph G(D′+C′, Ỹ) contracts to the underlying weighted graph of G_{(−m)} ⊖ T′_{1}; by 5.15 again, this contraction gives a birational morphism π′_{2} : Ỹ → S′, where S′ is smooth. Consider the divisor M′ = π′_{2}(D′ + C′) of S′ (with strong normal crossings); then G(M′, S′) is the underlying weighted graph of G_{(−m)} ⊖ T′_{1}, i.e., (G(M′, S′), Z′) = G_{(−m)} ⊖ T′_{1} for some component Z′ of M′. By 5.7, G_{(−m)} ⊖ T′_{1} is pseudo-linear, has at most two connected componenents, and the connected component which does not contain the distinguished vertex is an admissible chain. Thus we obtain an X-immersion (S′, μ′), where μ′ : S′ \ supp(M′) → X_{s} \ supp(F) is the isomorphism determined by π′_{2}, ω′, ω and X̃ \ supp(D + C) ≅ X_{s} \ supp(F). The X-immersion (S′, μ′) determines an affine ruling Λ′ of X and an element F′ of Λ′_{∗} satisfying supp(F′) = supp(F) (because the image of μ′ is X_{s} \ supp(F)). Also, Z′ is the 0-component of (S′, μ′) and let Σ′ be the section of (S′, μ′). Since the unique neighbor of the distinguished vertex of G_{(−m)} ⊖ T′_{1} has weight −m, we have (Σ′)^{2} = −m. Note that (X, Λ′)^{∼} = (Ỹ , |Z′|^{∼}), where |Z′|^{∼} denotes the strict transform of |Z′|. Also, \overline{HN}(π′_{2}, M′) is defined and is equal to T′_{2}.

Let π′_{1} : S′ → 𝔽_{m} be the unique birational morphism which contracts each reducible member of |Z′| to a 0-curve and whose exceptional locus is disjoint from Σ′ (1.2).

We claim that N(π′_{1}) = m′ − 2, where m′ is the number of irreducible components of M′ and N(π′_{1}) is (as usual) the number of irreducible components in the exceptional locus of π′_{1}. To see this, let π be the composition Ω →^{ω} X̃ →^{π_{2}} S →^{π_{1}} 𝔽_{n}, where π_{1} and π_{2} are the two morphisms determined by (X, Λ, F) as in 5.1. Then N(π) = |P| − 2 and

consequently N(π′) = |P| − 2, where π′ is the composition Ω →^{ω′} Ỹ →^{π′_{2}} S′ →^{π′_{1}} 𝔽_{m}. Since ω′ and π′_{2} correspond to contractions of graphs, it follows that N(π′_{1}) = |G_{(−m)} ⊖T′_{1}|−2, from which the claim follows.

Note that π′_{1}(Σ′) is the negative section of the standard ruling Λ_{m} of 𝔽_{m} and that π′_{1}(Z′) is a member of Λ_{m}. We claim that, for some member L of Λ_{m} other than π′_{1}(Z′),

(30) π′_{1}(M′) ⊆ L + π′_{1}(Σ′) + π′_{1}(Z′), center(π′_{1}) ⊂ L \ π′_{1}(Σ′) and \overline{HN}(π′_{1}, L) = T′_{1}.

The verification of this splits into two cases.

If m′ = 2 then |G_{(−m)} ⊖ T′_{1}| = 2, so T′_{1} is the empty tableau. On the other hand, N(π′_{1}) = m′ − 2 = 0 implies that π′_{1} is an isomorphism. If we let L be any member of Λ_{m} other than π′_{1}(Z′), then (30) holds.

If m′ > 2 then G_{(−m)} ⊖T′_{1} has more than 2 vertices, so the middle vertex of G_{(−m)} has exactly two neighbors, i.e., Σ′ has two neighbors Z′ and Z′′ in M′. Let M′_{1}, …, M′_{m′−3} be the components of M′ other than Z′, Σ′ and Z′′; since each M′_{i} is contained in a member of the ruling |Z′| (because M′_{i} ∩ Z′ = ∅) and is disjoint from Σ′, each M′_{i} is shrunk by π′_{1}. Since N(π′_{1}) = m′ −2, the exceptional locus of π′_{1} is E ∪M′_{1} ∪⋯∪M′_{m′−3}, for some curve E not contained in M′_{1} ∪ ⋯ ∪ M′_{m′−3}. Since (M′_{i})^{2} < −1 for all i, E is the unique (−1)-component of the exceptional locus of π′_{1}. Let L = π′_{1}(Z′′) and note that L is a member of Λ_{m} other than π′_{1}(Z′) and satisfying:

π′_{1}(M′) = L + π′_{1}(Σ′) + π′_{1}(Z′) and center(π′_{1}) ⊂ L \ π′_{1}(Σ′).

Using G_{(−m)} = (G(π′_{1}(M′), 𝔽_{m}), L, π′_{1}(Z′)), we obtain

G_{(−m)} ⊖ \overline{HN}(π′_{1}, L) = (G(π′_{1}(M′), 𝔽_{m}), L, π′_{1}(Z′)) ⊖ \overline{HN}(π′_{1}, L)
= (G(M′, S′), Z′) = G_{(−m)} ⊖ T′_{1},

so \overline{HN}(π′_{1}, L) = T′_{1} by 5.14 and (30) holds in this case too.

We conclude that π′_{1} and π′_{2} are the two morphisms determined by (X, Λ′, F′) (5.1) and that disc(X, Λ′, F′) = (m, T′_{1}, T′_{2}). □

The following order relation is useful for describing 𝕋(X) explicitely: Definition 5.16. We define a transitive relation > on the set 𝕋 by declaring that (n, T_{1}, T_{2}) > (m, T′_{1}, T′_{2}) if n = 1 and the following holds:

Let L be the weighted pair such that G_{(−m)}⊖T′_{1} is pseudo-linear of type (m−1, L). Then there exist an integer s ≥ 1 and tableaux X_{1}, …, X_{s} such that T_{1} = (T′_{1})^{(ˇ s)}, T_{2} = X_{s} ⋯ X_{1}T′_{2} and X_{i} ∈ T_{k_{i}}(L^{t^{i}}), where k_{1} = m − 1 and k_{i} = 0 for all i > 1.

We define the symbols <, ≥ and ≤ the usual way. (See Definition 3.21 for T_{k}(L).) Remark. There cannot be an infinite descending sequence τ_{1} > τ_{2} > ⋯ in 𝕋. Indeed, if (n, T_{1}, T_{2}) > (m, T′_{1}, T′_{2}) then the number of columns of T′_{2} is strictly less than that of T_{2}.

Note that, if τ′ ∈ 𝕋 is given, we may explicitely describe all τ ∈ 𝕋 satisfying τ > τ′ (this is done in 5.39, below). Thus the following (see also Corollary 5.22) describes 𝕋(X) in terms of 𝕋_{0}(X): Corollary 5.17. Let X be a surface satisfying (†).

(1) If τ, τ′ ∈ 𝕋 are such that τ > τ′, then τ ∈ 𝕋(X) ⟺ τ′ ∈ 𝕋(X).

(2) Given any τ ∈ 𝕋(X) \ 𝕋_{0}(X), there exists τ′ ∈ 𝕋_{0}(X) such that τ > τ′.

Although 5.17 is essentially a corollary of Theorem 5.13, its proof requires some preparation. Lemma 5.18. Let (n, T_{1}, T_{2}), (m, T′_{1}, T′_{2}) ∈ 𝕋.

(1) If (n, T_{1}, T_{2}) > (m, T′_{1}, T′_{2}) then (n, T_{1}, T_{2}) ≡ (m, T′_{1}, T′_{2}).

(2) If (n, T_{1}, T_{2}) ≡ (m, T′_{1}, T′_{2}) then (G_{(−n)} ⊖ T_{1}) ⊖ T_{2} and (G_{(−m)} ⊖ T′_{1}) ⊖ T′_{2} are equivalent weighted graphs and consequently

(n, T_{1}, T_{2}) ∈ 𝕋(†) ⟺ (m, T′_{1}, T′_{2}) ∈ 𝕋(†).

Proof. Suppose that (n, T_{1}, T_{2}) > (m, T′_{1}, T′_{2}). Recall that n = 1 and let the notations (L, s, X_{1}, …, X_{s}, k_{i}) be as in Definition 5.16. Since G_{(−m)} ⊖ T′_{1} is pseudo-linear of type (m − 1, L), it follows that G_{(−1)} ⊖ T′_{1} is pseudo-linear of type (0, L), so G_{(−1)} ⊖ (T′_{1})^{(ˇ s)} = (G_{(−1)} ⊖ T′_{1})^{t^{s}} is pseudo-linear of type (0, L^{t^{s}}). By Corollary 3.37, (G_{(−1)} ⊖ (T′_{1})^{(ˇ s)})X_{s} ⋯ X_{1} is pseudo-contractible of type (m − 1, L).

Let P ≍ P′ mean, temporarily, that the weighted pairs P and P′ are the same outside of the connected component containing the distinguished vertex. Then

G_{(−m)} ⊖ T′_{1} ≍ G_{(−1)} ⊖ T′_{1} ≍ G_{(−1)} ⊖ (T′_{1})^{(ˇ s)} ≍ (G_{(−1)} ⊖ (T′_{1})^{(ˇ s)})X_{s} ⋯ X_{1},

where the second “≍” follows from part (2) of 5.14 and the other two are obvious. Thus the weighted pairs (G_{(−1)} ⊖ (T′_{1})^{(ˇ s)})X_{s} ⋯ X_{1} and G_{(−m)} ⊖ T′_{1} are pseudo-contractible of the same type, and identical outside of the connected component containing the distinguished vertex; it follows that

(31)  (G_{(−1)} ⊖ (T′_{1})^{(ˇ s)})X_{s} ⋯ X_{1} ≈ G_{(−m)} ⊖ T′_{1}

and consequently

(G_{(−1)} ⊖ T_{1})T_{2} = (G_{(−1)} ⊖ (T′_{1})^{(ˇ s)})X_{s} ⋯ X_{1}T′_{2} ≈ (G_{(−m)} ⊖ T′_{1})T′_{2},

which proves assertion (1). If (n, T_{1}, T_{2}) ≡ (m, T′_{1}, T′_{2}) then (G_{(−n)} ⊖ T_{1})T_{2} ≈ (G_{(−m)} ⊖ T′_{1})T′_{2}, so

(G_{(−n)} ⊖ T_{1}) ⊖ T_{2} = (G_{(−n)} ⊖ T_{1})T_{2} ⊖ 1 ∼ (G_{(−m)} ⊖ T′_{1})T′_{2} ⊖ 1 = (G_{(−m)} ⊖ T′_{1}) ⊖ T′_{2}

and (2) holds. □

Lemma 5.19. Let τ = (n, T_{1}, T_{2}) ∈ 𝕋(†). If the weighted graph (G_{(−n)} ⊖ T_{1}) ⊖ T_{2} can be contracted to a weighted graph whose number of branch points is strictly less than that of (G_{(−n)} ⊖ T_{1}) ⊖ T_{2}, then τ > τ′ for some τ′ ∈ 𝕋(†).

Proof. Let L be the weighted pair such that P = G_{(−n)} ⊖ T_{1} is pseudo-linear of type (n − 1, L^{t}). Note that T_{2} ∈ T^{#} but that, since (G_{(−n)} ⊖ T_{1}) ⊖ T_{2} has a branch point, T_{2} satisfies none of the conditions of 5.6 (this follows from 5.7). Thus, if we write T_{2} = CT with C, T ∈ T and C a single column, we have C ≠ \binom{1}{1}, T ≠ 1 and if T is a single column then it is not of the form \binom{1}{k}. Consider the weighted pair P C = (H, e) and regard e as a vertex of (G_{(−n)} ⊖ T_{1}) ⊖ T_{2} = (H, e) ⊖ T. Then (G_{(−n)} ⊖ T_{1}) ⊖ T_{2} has three branches at e, say B, B′ and B′′, where B contains the vertices of P, B ∪ B′ contains no branch point of (G_{(−n)} ⊖ T_{1}) ⊖ T_{2} and every weight in B ∪ B′ ∪ B′′ is strictly less than −1, except possibly the middle vertex of G_{(−n)} (which belongs to B and has weight −n). Since (G_{(−n)} ⊖T_{1})⊖T_{2} contracts to a graph with less branch points, n = 1 (so P is of type (0, L^{t})) and B shrinks. In other words, the connected component L^{t}C of P C (regard L^{t}C as a weighted pair) contracts to a linear pair. By 3.32, L^{t}C\binom{1}{1}^{ν} is contractible of type L and C\binom{1}{1}^{ν} ∈ T (L^{t}), for some ν ∈ ℕ; so P C is pseudo-contractible of type (ν, L).

We may write T_{2} = X_{1}T′_{2} with X_{1} = C\binom{1}{1}^{ℓ} ∈ T (some ℓ ∈ ℕ) and T′_{2} ∈ T^{#}. If ℓ > ν then P X_{1} contracts to a weighted pair (W, w) which contains a vertex v ≠ w of nonnegative weight; then (G_{(−1)} ⊖ T_{1}) ⊖ T_{2} = P X_{1} ⊖ T′_{2} contracts to a weighted graph containing a nonnegative weight, contradicting the fact that its intersection matrix is negative definite. So ℓ ≤ ν and consequently X_{1} ∈ T_{ν−ℓ}(L^{t}), which we rewrite as X_{1} ∈ T_{m−1}(L^{t}), where m ≥ 1. It is then clear that the triple τ′ = (m, Ť_{1}, T′_{2}) belongs to 𝕋 and satisfies τ > τ′. By 5.18, τ′ ∈ 𝕋(†). □

Proof of Corollary 5.17. Since τ > τ′ implies τ ≡ τ′ by Lemma 5.18, assertion (1) of 5.17 follows from 5.13. Also, (2) follows from (1): If τ ∈ 𝕋(X)\𝕋_{0}(X) then Lemma 5.19 implies that τ is not minimal in (𝕋(†), <); since there is no infinite descending sequence in (𝕋(†), <), we may therefore choose a minimal τ′ in (𝕋(†), <) such that τ > τ′; then (1) implies τ′ ∈ 𝕋(X), so τ′ ∈ 𝕋_{0}(X) by Lemma 5.19. □

Notation 5.20. (1) Given τ ∈ 𝕋, define

[τ, ∞) = {τ′ ∈ 𝕋 | τ′ ≥ τ} and (−∞, τ] = {τ′ ∈ 𝕋 | τ′ ≤ τ} .

(2) If X satisfies (†), min(𝕋(X)) = {τ ∈ 𝕋(X) | τ is a minimal element of (𝕋(†), <)}. Remark. By Lemma 5.19, min(𝕋(X)) ⊆ 𝕋_{0}(X). Lemma 5.21. Given τ ∈ 𝕋, the set (−∞, τ] is totally ordered and finite. Consequently, if τ is not minimal in 𝕋 then there exists exactly one τ^{−} ∈ 𝕋 such that

τ > τ^{−} and no τ^{∗} ∈ 𝕋 satisfies τ > τ^{∗} > τ^{−}.

We call τ^{−} the immediate predecessor of τ.

Proof. We show that if τ, τ′, τ′′ ∈ 𝕋 satisfy τ > τ′ and τ > τ′′, then τ′ ≥ τ′′ or τ′ ≤ τ′′. Write τ = (1, T_{1}, T_{2}), τ′ = (m′, T′_{1}, T′_{2}) and τ′′ = (m′′, T′′_{1}, T′′_{2}) and let q′ (resp. q′′) be the number of columns of T′_{2} (resp. T′′_{2}). We may assume that q′ ≤ q′′.

Since τ > τ′, we have T_{2} = X_{s} ⋯ X_{1}T′_{2} (notation as in Definition 5.16) and similarly τ > τ′′ gives T_{2} = Y_{r} ⋯ Y_{1}T′′_{2}. Thus T′_{2} (resp. T′′_{2}) consists of the rightmost q′ (resp. q′′) columns of T_{2}; since q′ ≤ q′′, it follows that T′′_{2} = W T′_{2} for some W ∈ T (and T′′_{2} ∈ T^{#} implies W ∈ T^{#}). So X_{s} ⋯ X_{1}T′_{2} = T_{2} = Y_{r} ⋯ Y_{1}W T′_{2} and consequently X_{s} ⋯ X_{1} = Y_{r} ⋯ Y_{1}W. Since the X_{i} are irreducible elements of the monoid T^{#}, it follows that W = X_{j} ⋯ X_{1} (some j ≥ 0) by unique factorization in T^{#} (see 3.34). Thus T′′_{2} = X_{j} ⋯ X_{1}T′_{2} and it follows that τ′′ > τ′ or τ′′ = τ′. This shows that (−∞, τ] is totally ordered; the other assertions are trivial. □

Corollary 5.22. If X satisfies (†) then {[τ, ∞) | τ ∈ min(𝕋(X))} is a partition of 𝕋(X). Remark. [τ, ∞) is described explicitely in 5.39, below.

Proof. By Corollary 5.17, the union of the sets [τ, ∞) is 𝕋(X). If τ′, τ′′ ∈ min(𝕋(X)) are such that [τ′, ∞) ∩ [τ′′, ∞) ≠ ∅, then Lemma 5.21 implies τ′ = τ′′. □

In relation with the reduction process of Corollary 2.11, we give: Proposition 5.23. Let X be a surface satisfying (†), Λ a non-basic affine ruling of X and F the unique element of Λ_{∗}. Consider the pair (Λ^{−}, F^{−}) obtained from (Λ, F) by means of the reduction process of Corollary 2.11, i.e., if I = (S, μ) is the distinguished X-immersion determining (Λ, F), P ∈ S the center of the morphism π_{2} : X̃ → S which contracts F̃ to a 0-curve and π any^{8} element of Π_{P} (I), then (Λ^{−}, F^{−}) is the pair determined by the X-immersion I ∗ π. Let τ and τ^{−} denote the discrete parts of (Λ, F) and (Λ^{−}, F^{−}) respectively. Then τ^{−} is the immediate predecessor of τ (see 5.21).

Proof. Write τ = (n, T_{1}, T_{2}). Since Λ is non-basic, (G_{(−n)} ⊖ T_{1}) ⊖ T_{2} ≅ G(Λ) can be contracted to a weighted graph with less branch points; then the proof of Lemma 5.19 produces a τ′ = (m, T′_{1}, T′_{2}) such that τ > τ′, T_{2} = X_{1}T′_{2} and X_{1} = C\binom{1}{1}^{ℓ} (note that τ′ is the immediate predecessor of τ). Since τ > τ′ implies τ ≡ τ′, the proof of Theorem 5.13 produces a pair (Λ′, F′) whose discrete part is τ′. The factorization T_{2} = C\binom{1}{1}^{ℓ}T′_{2} determines a factorization of π_{2} as

(32)  X̃ ⟶_{T′_{2}} R′′ \overrightarrow{\binom{1}{1}^{ℓ}} S̃ ⟶_{C}^{π} S

and C ∈ T_{ν}(L^{t}) implies that π ∈ Π_{P} (S, μ). Then we see that the X-immersion (S′, μ′) (in the proof of Theorem 5.13) is equivalent to I ∗ π. Since (S′, μ′) determines (Λ′, F′) and I ∗ π determines (Λ^{−}, F^{−}), this means that (Λ′, F′) = (Λ^{−}, F^{−}). So τ′ = τ^{−} and we are done. □

Surfaces satisfying (‡)

Notation 5.24. (1) Consider triples (X, Λ, F) where X satisfies (†), Λ is an affine ruling of X and F ∈ Λ_{∗}. Two such triples are equivalent, (X, Λ, F) ∼ (X′, Λ′, F′), when there exists an isomorphism X → X′ which transforms Λ into Λ′ and F into F′. If this is the case then disc(X, Λ, F) = disc(X′, Λ′, F′), so we may speak of the discrete part of the equivalence class [X, Λ, F] of (X, Λ, F). So we obtain a map disc : 𝕊(†) → 𝕋(†), where 𝕊(†) denotes the set of equivalence classes [X, Λ, F].

8For the fact that (Λ^{−}, F^{−}) is independent of the choice of π ∈ Π_{P} (I), see the last assertion of Corollary 4.4.

(2) We will also consider the restriction disc : 𝕊_{0}(‡) → 𝕋_{0}(‡) of the above map 𝕊(†) → 𝕋(†), where 𝕊_{0}(‡) = {[X, Λ, F] ∈ 𝕊(†) | X satisfies (‡) and Λ is basic} and where 𝕋_{0}(‡) is the set of (m, T_{1}, T_{2}) ∈ 𝕋 such that (i) each of T_{1}, T_{2} satisfies one of conditions (1–3) of 5.6; and (ii) the weighted graph (G_{(−m)} ⊖ T_{1}) ⊖ T_{2} has a negative definite intersection matrix. (See Lemma 5.11 for the fact that disc maps 𝕊_{0}(‡) in 𝕋_{0}(‡); see also 5.41.)

(3) Let \overline{𝕊}(‡) be the set of isomorphism classes of surfaces satisfying (‡). The isomorphism class of X is denoted [X]. Then [X, Λ, F] ↦ [X] defines a map 𝕊_{0}(‡) → \overline{𝕊}(‡).

In particular, we will show: Proposition 5.25. 𝕊_{0}(‡) → \overline{𝕊}(‡) and 𝕊(†) → 𝕋(†) are surjective and 𝕊_{0}(‡) → 𝕋_{0}(‡) is bijective.

Proof that 𝕊_{0}(‡) → \overline{𝕊}(‡) is surjective. If X is any surface satisfying (‡), then X admits a basic affine ruling Λ by Theorem 2.1 and Λ_{∗} ≠ ∅ by 2.5; thus [X] is in the image of 𝕊_{0}(‡) → \overline{𝕊}(‡). □

The proof of the other assertions requires some preparation. Definition 5.26. Let m be a positive integer, Λ_{m} the standard ruling of 𝔽_{m} and Σ_{m} ⊂ 𝔽_{m} the negative section of Λ_{m} (Σ_{m}^{2} = −m). Let T_{1}, T_{2} ∈ T .

(1) By a blowing-up of 𝔽_{m} according to (T_{1}, T_{2}), we mean a triple (π, P_{1}, P_{2}) where π : Y → 𝔽_{m} is a birational morphism (with Y smooth and complete), P_{1}, P_{2} are points of 𝔽_{m} \ Σ_{m} belonging to distinct members of Λ_{m} (P_{i} ∈ Z_{i} ∈ Λ_{m}, Z_{1} ≠ Z_{2}), center(π) ⊆ {P_{1}, P_{2}} and, for each i = 1, 2, π^{−1}(P_{i}) contains at most one (−1)-curve and \overline{HN}(π, Z_{i}) = T_{i}.

(2) Let β = (π, P_{1}, P_{2}) and β′ = (π′, P′_{1}, P′_{2}) be two blowings-up of 𝔽_{m} according to (T_{1}, T_{2}). We say that β is equivalent to β′ if there exists a commutative diagram:

Y ⟶^{≅} Y′
↓^{π}  ↓_{π′}
𝔽_{m} ⟶_{φ}^{≅} 𝔽_{m}

where the horizontal arrows are isomorphisms and, for each i = 1, 2, φ(P_{i}) = P′_{i}. Lemma 5.27. Let (m, T_{1}, T_{2}) ∈ ℤ^{+} × T × T be such that:

(i) Each T_{i} satisfies one of conditions (1–3) of 5.6; and (ii) if both T_{i} are nonempty then mc_{1}c_{2} − c_{1}p_{2} − c_{2}p_{1} ≠ 0, where \binom{p_{i}}{c_{i}} is the first

column of T_{i}. Then any two blowings-up of 𝔽_{m} according to (T_{1}, T_{2}) are equivalent.

Proof. Let β = (π, P_{1}, P_{2}) and β′ = (π′, P′_{1}, P′_{2}) be two blowings-up of 𝔽_{m} according to (T_{1}, T_{2}). Since there exists an automorphism^{9} of 𝔽_{m} which maps P_{1} and P_{2} to P′_{1} and P′_{2} respectively, we may assume that (P_{1}, P_{2}) = (P′_{1}, P′_{2}). Let Z_{i} be the member of Λ_{m} containing P_{i} (Z_{1} ≠ Z_{2}) and choose a section S of Λ_{m} such that S ∩ Σ_{m} = ∅ and P_{1}, P_{2} ∈ S. We can write 𝔽_{m} \ Σ_{m} = Spec k[x_{1}, y_{1}] ∪ Spec k[x_{2}, y_{2}], where x_{i}, y_{i} are local equations at P_{i} for Z_{i} and S respectively, x_{2} = x_{1}^{−1} and y_{2} = y_{1}x_{1}^{−m}. Then the Hamburger-Noether tableaux HN_{i} = HN(π; x_{i}, y_{i}) and HN′_{i} = HN(π′; x_{i}, y_{i}) satisfy \overline{HN}_{i} = T_{i} = \overline{HN}′_{i} (i = 1, 2).

Note that, for each (σ, τ) ∈ (k^{∗})^{2}, x_{1} ↦ σx_{1}, y_{1} ↦ τy_{1} induces an automorphism φ_{σ,τ} of 𝔽_{m} which leaves (P_{1}, P_{2}, S) unchanged.

Let I = {i ∈ {1, 2} | T_{i} has two columns} and, for each i ∈ I, define α_{i}, α′_{i} ∈ k^{∗} by saying that ( \begin{matrix}∗ \\ ∗ \\ α_{i} \end{matrix} ) (resp. ( \begin{matrix}∗ \\ ∗ \\ α′_{i} \end{matrix} )) is the first column of HN_{i} (resp. HN′_{i}). If the two sequences {α_{i}}_{i∈I} and {α′_{i}}_{i∈I} are equal, then the assertion is trivial. So it suffices to show that the sequence {α_{i}}_{i∈I} can be transformed into the constant sequence with value 1 by composing π with automorphisms φ_{σ,τ}.

Let us study the following situation. Let P_{i}, Z_{i}, S and (x_{i}, y_{i}) be as above. Let α_{1}, α_{2} ∈ k^{∗} and p_{1}, c_{1}, p_{2}, c_{2} ∈ ℕ be such that 0 < p_{i} ≤ c_{i} are relatively prime (i = 1, 2) and mc_{1}c_{2} − c_{1}p_{2} − c_{2}p_{1} ≠ 0; consider a birational morphism f : Y → 𝔽_{m} (Y smooth and complete) satisfying center(f) = {P_{1}, P_{2}} and, for each i = 1, 2, f^{−1}(P_{i}) contains a unique (−1)-curve E_{i} and HN(f; x_{i}, y_{i}) = ( \begin{matrix}p_{i} \\ c_{i} \\ α_{i} \end{matrix} ). For each i = 1, 2, the HN-algorithm of [11] produces a parameter u_{i} for E_{i} ≅ ℙ^{1} and the condition u_{i} = α_{i} determines a point on E_{i}. Moreover, u_{i} = y_{i}^{c_{i}}/x_{i}^{p_{i}} or u_{i} = x_{i}^{p_{i}}/y_{i}^{c_{i}}. We have φ_{σ,τ}(u_{i}) = σ^{ν_{i}}τ^{μ_{i}}u_{i}, with

(ν_{1}, μ_{1}) = ±(−p_{1}, c_{1}) and (ν_{2}, μ_{2}) = ±(p_{2} − mc_{2}, c_{2}). Since | \begin{matrix}ν_{1} & μ_{1} \\ ν_{2} & μ_{2} \end{matrix} | = ±(mc_{1}c_{2} − c_{1}p_{2} −

c_{2}p_{1}) ≠ 0, we may choose (σ, τ) such that

HN(φ_{σ,τ} ∘ f; x_{1}, y_{1}) = ( \begin{matrix}p_{1} \\ c_{1} \\ 1 \end{matrix} ) and HN(φ_{σ,τ} ∘ f; x_{2}, y_{2}) = ( \begin{matrix}p_{2} \\ c_{2} \\ 1 \end{matrix} ) .

□

Lemma 5.28. Let (m, T_{1}, T_{2}) ∈ 𝕋_{0}(‡).

(1) If both T_{i} are nonempty then mc_{1}c_{2} − c_{1}p_{2} − c_{2}p_{1} > 0, where \binom{p_{i}}{c_{i}} is the first column of T_{i}.

(2) The blowing-up of 𝔽_{m} according to (T_{1}, T_{2}) is unique, up to equivalence.

Proof. If both T_{i} are nonempty then let Γ be the connected component of

(G_{(−m)} ⊖ \binom{p_{1}}{c_{1}}) ⊖ \binom{p_{2}}{c_{2}}

containing the vertices of G_{(−m)}. Since Γ is a subgraph of (G_{(−m)} ⊖ T_{1}) ⊖ T_{2}, it must have a negative definite intersection matrix. In particular, det(Γ) > 0. By 3.16 and Lemma 3.23, det(Γ) = mc_{1}c_{2} − c_{1}p_{2} − c_{2}p_{1}. This proves (1), and (2) follows from (1) and Lemma 5.27. □

9The automorphism preserves fibres and Σ_{m}, since m > 0.

5.29 (Proof of Proposition 5.25, continued). Given τ = (m, T_{1}, T_{2}) ∈ 𝕋(†), we describe a method for constructing all (X, Λ, F) such that disc(X, Λ, F) = τ (where X satisfies (†), Λ is an affine ruling of X and F ∈ Λ_{∗}). This will show, in particular, that disc : 𝕊(†) → 𝕋(†) is surjective.

Choose a blowing-up

(X̃ →^{π} 𝔽_{m}, P_{1}, P_{2})

of 𝔽_{m} according to (T_{1}, T_{2}) and let Z_{1} and Z_{2} be the elements of Λ_{m} satisfying P_{i} ∈ Z_{i}. Recall that center(π) ⊆ \{P_{1}, P_{2}\} and \overline{HN}(π, Z_{i}) = T_{i}. For i = 1, 2, define

E_{i} = \{ \begin{matrix}π^{−1}(Z_{i}), & if P_{i} ∉ center(π), \\ the (−1)-curve in π^{−1}(P_{i}), & if P_{i} ∈ center(π) \end{matrix}

and let D be the divisor of X̃ with strong normal crossings defined by π^{−1}(supp(Z_{1} + Σ_{m}+Z_{2})) = supp(E_{1}+D+E_{2}) and E_{1}, E_{2} ⊈ supp(D). Then G(D, X̃) = (G_{(−m)}⊖T_{1})⊖ T_{2} and consequently D has a negative definite intersection matrix (because τ ∈ 𝕋(†)). So there exists a complete normal surface X and a birational morphism X̃ → X with exceptional locus supp(D). Note that Λ_{m} determines an affine ruling Λ of X, because 𝔽_{m} ← X̃ → X restrict to an isomorphism between 𝔽_{m} \ supp(Z_{1} + Σ_{m} + Z_{2}) and an open subset of X. Moreover, if Λ̃ is the strict transform of Λ_{m} with respect to π, then (X̃, Λ̃) = (X, Λ)^{∼}. Equation (4) of 1.7 implies that Pic(X_{s}) has rank 1, so X satisfies (†). Note that the image of E_{i} under X̃ → X is the support of some F_{i} ∈ Λ; moreover, F_{2} ∈ Λ_{∗} and disc(X, Λ, F_{2}) = τ. It is clear, also, that (X, Λ, F_{2}) is determined by the choice of the blowing-up (π, P_{1}, P_{2}) and that every triple (X, Λ, F) with discrete part τ can be obtained in this way, i.e., by choosing a suitable blowing-up. □ 5.30 (End of proof of Proposition 5.25). We show that 𝕊_{0}(‡) → 𝕋_{0}(‡) is bijective. Given τ = (m, T_{1}, T_{2}) ∈ 𝕋_{0}(‡), consider a triple (X, Λ, F_{2}) constructed as in 5.29. By Lemma 5.11, X satisfies (‡) and Λ is basic, so [X, Λ, F_{2}] ∈ 𝕊_{0}(‡). Also, uniqueness (5.28) of the blowing-up (π, P_{1}, P_{2}) up to equivalence implies uniqueness of (X, Λ, F_{2}) up to equivalence; in other words, τ ↦ [X, Λ, F_{2}] is a well-defined map 𝕋_{0}(‡) → 𝕊_{0}(‡), and this is the inverse of the “discrete part” map 𝕊_{0}(‡) → 𝕋_{0}(‡). □ Corollary 5.31. There exists a surjective map f : 𝕋_{0}(‡) → \overline{𝕊}(‡) satisfying:

Given τ ∈ 𝕋_{0}(‡) and X satisfying (‡), f(τ) = [X] if and only if there exists an affine ruling Λ of X and an F ∈ Λ_{∗} such that τ is the discrete part of (Λ, F).

Remark. One interesting aspect of the surjection f : 𝕋_{0}(‡) → \overline{𝕊}(‡) of Corollary 5.31 is that, given τ ∈ 𝕋_{0}(‡), we may construct, in a very explicit way, a surface X such that f(τ) = [X] (the construction is carried out in 5.29). Since the elements of 𝕋_{0}(‡) can be described explicitely (5.41), this gives an interesting description of the class of surfaces satisfying (‡). Corollary 5.32. Let X_{1} and X_{2} be surfaces satisfying (†) and such that 𝕋(X_{1}) ∩ 𝕋(X_{2}) ≠ ∅. Then:

(1) 𝕋_{0}(X_{1}) ∩ 𝕋_{0}(X_{2}) ≠ ∅.

(2) If at least one of X_{1}, X_{2} satisfies (‡), then X_{1} ≅ X_{2}.

Proof. Assertion (1) follows immediately from Corollary 5.17. To prove (2), assume that X_{1} satisfies (‡) and consider τ = (m, T_{1}, T_{2}) ∈ 𝕋_{0}(X_{1}) ∩ 𝕋_{0}(X_{2}). By Lemma 5.11, each of T_{1}, T_{2} satisfies one of the conditions (1–3) of 5.6; since τ ∈ 𝕋(X_{2}), 5.11 implies that X_{2} satisfies (‡). Then the surjection f : 𝕋_{0}(‡) → \overline{𝕊}(‡) of Corollary 5.31 satisfies f(τ) = [X_{1}] and f(τ) = [X_{2}], so [X_{1}] = [X_{2}]. □

5.33. Consider the equivalence relation “∼” on 𝕋 which is generated by declaring that τ ∼ τ′ whenever τ < τ′. Then τ ∼ τ′ ⟹ τ ≡ τ′, but the converse does not hold. Indeed, 5.21 implies that 𝕋/∼ = {[τ, ∞) | τ ∈ min(𝕋)}, so each equivalence class with respect to ∼ contains exactly one minimal element of (𝕋, <). However, if τ = (1, ( \begin{matrix}16 & 1 \\ 39 & 3 \end{matrix} ) , ( \begin{matrix}135 & 1 \\ 229 & 4 \end{matrix} )) and τ′ = (1, ( \begin{matrix}23 & 1 \\ 39 & 3 \end{matrix} ) , ( \begin{matrix}2 & 1 \\ 5 & 4 \end{matrix} )) then τ and τ′ are distinct minimal elements of (𝕋, <) and τ ≡ τ′.

Regarding the relation ∼ of 5.33, we have the following: Corollary 5.34. For i = 1, 2, let X_{i} be a surface satisfying (‡), let Λ_{i} be an affine ruling of X_{i} and let F_{i} ∈ (Λ_{i})_{∗}. If disc(X_{1}, Λ_{1}, F_{1}) ∼ disc(X_{2}, Λ_{2}, F_{2}), then there exist (Λ′_{1}, F′_{1}) and (Λ′_{2}, F′_{2}) satisfying:

(1) For each i, Λ′_{i} is a basic affine ruling of X_{i}, F′_{i} ∈ (Λ′_{i})_{∗} and supp(F′_{i}) = supp(F_{i});

(2) there exists an isomorphism X_{1} → X_{2} which carries Λ′_{1} to Λ′_{2} and F′_{1} to F′_{2}. In particular, there exists an isomorphism X_{1} → X_{2} which maps supp(F_{1}) onto supp(F_{2}).

Proof. Let τ_{i} ∈ 𝕋 be the discrete part of (Λ_{i}, F_{i}). Then Lemma 5.21 implies that there exists τ′ ∈ 𝕋 such that (for all i) τ_{i} ≥ τ′; clearly, τ′ may be chosen so that it is a minimal element of 𝕋. By Theorem 5.13, for each i there exists an affine ruling Λ′_{i} of X_{i} and F′_{i} ∈ (Λ′_{i})_{∗} satisfying supp(F′_{i}) = supp(F_{i}) and such that the discrete part of (X_{i}, Λ′_{i}, F′_{i}) is τ′. So (5.19) Λ′_{i} is basic and the two elements [X_{1}, Λ′_{1}, F′_{1}] and [X_{2}, Λ′_{2}, F′_{2}] of 𝕊_{0}(‡) have the same image (namely, τ′) under the bijective map 𝕊_{0}(‡) → 𝕋_{0}(‡). Hence, [X_{1}, Λ′_{1}, F′_{1}] = [X_{2}, Λ′_{2}, F′_{2}]. □

Multiplicities

Definition 5.35. Given a tableau T = ( \begin{matrix}p_{1} & ··· & p_{k} \\ c_{1} & ··· & c_{k} \end{matrix}) ∈ T , we define

μ(T) = \{ \begin{matrix}1, & if T = 1, \\ c_{1} ⋯ c_{k}, & else. \end{matrix}

Note that μ : T → ℕ \ {0} is a homomorphism of multiplicative monoids.

Remark. Given a finite Hamburger-Noether tableau HN = ( \begin{matrix}p_{1} & ··· \\ c_{1} & ··· \\ α_{1} & ··· \end{matrix} ), consider T = \overline{HN} ∈ T defined as in 3.6. Then μ(T) = c_{1} (or 1, if HN is empty).

By the above remark and A.10.1 of [11], we have 5.36. Let f : X → Y be a birational morphism of smooth complete surfaces and D a nonzero divisor of Y with strong normal crossings. Assume that the exceptional locus of f contains at most one (−1)-curve and that the center of f, if nonempty, is a point P belonging to exactly one component Z of D. Let

E = \{ \begin{matrix}f^{−1}(Z), & if f is an isomorphism, \\ the (−1)-curve in f^{−1}(P), & if f is not an isomorphism. \end{matrix}

Then the multiplicity of E in the total transform of D is equal to μ(\overline{HN}(f, D)). The above statement and Proposition 1.8 give:

Corollary 5.37. Let X be a surface satisfying (†). If (m, T_{1}, T_{2}) is the discrete part of (Λ, F), where Λ is an affine ruling of X and F ∈ Λ_{∗}, and if G ∈ Λ \ {F } is such that {F, G} contains all multiple members of Λ (such a G exists, by definition of Λ_{∗}), then

F = μ(T_{2})C_{2} and G = μ(T_{1})C_{1},

where C_{1}, C_{2} ⊂ X are (irreducible) curves. Moreover, Pic(X_{s}) ≅ ℤ ⊕ ℤ/dℤ, where d = gcd(μ(T_{1}), μ(T_{2})). Remark. If X = ℙ^{2}, or more generally a weighted projective plane ℙ(a, b, c) where a, b, c are pairwise relatively prime, then μ(T_{1}) = deg C_{2} and μ(T_{2}) = deg C_{1}. (In view of the above result, this follows immediately from gcd(deg C_{1}, deg C_{2}) = 1, for which we refer to [5] or [6].)

See also Corollary 5.40.

Some explicit computations

5.38. Let m > 0 be an integer and suppose that T ∈ T satisfies one of conditions (1–3) of 5.6.

(1) Recall that G_{(−m)} ⊖ T is pseudo-linear of type (m − 1, L), where L is a weighted pair satisfying the condition (0), uniquely determined by T. Then Lemma 3.23 gives:

M(L) = \{ \begin{matrix}( \begin{matrix}0 & 1 \\ 1 & 1 \end{matrix} ) & if T satisfies 5.6.1, \\ ( \begin{matrix}c−p−p′+p′′ & c−p \\ c−p′ & c \end{matrix} ) & if T satisfies 5.6.2, \\ ( \begin{matrix}Np(c−p)−1 & Nc^{2}−Ncp−1 \\ Ncp−1 & Nc^{2} \end{matrix} ) & if T satisfies 5.6.3, \end{matrix}

where p′ and p′′ are defined by \binom{p′′}{p′} = \binom{p}{c}^{∗}. Note that L is degenerate if and

only if T ∈ \{1, ( \begin{matrix}1 & 1 \\ 2 & 1 \end{matrix} )\} ∪ \{\binom{n}{n+1} | n ≥ 1\}.

(2) The conditions

M(L) = ( \begin{matrix}γ̇(T) & α_{1} \\ α_{2} & γ(T) \end{matrix} ) and α_{i+2} = α_{i} (i ≥ 1)

define positive integers γ̇(T) and γ(T) and an infinite sequence α(T) = (α_{1}, α_{2}, …) of positive integers. Note that these are uniquely determined by T and can be computed from (1). They satisfy M(L^{t^{i}}) = ( \begin{matrix}γ̇(T) & α_{i+1} \\ α_{i} & γ(T) \end{matrix} ) for all i ≥ 1.

(3) A sequence ν = (ν_{1}, …, ν_{s}) of natural numbers is said to be (m, T)-admissible if s ≥ 1 and the following conditions hold:

(a) If T ∈ \{1, ( \begin{matrix}1 & 1 \\ 2 & 1 \end{matrix} )\} ∪ \{\binom{n}{n+1} | n ≥ 1\}, ν_{1} ≥ max(1, m − 1) and ν_{i} ≥ 1 for all

i > 1.

(b) For all other T, ν_{1} ≥ m − 1 and ν_{i} ≥ 0 for all i > 1.

(4) Given an (m, T)-admissible sequence ν = (ν_{1}, …, ν_{s}), consider the sequence of tableaux (X_{1}, …, X_{s}) ∈ T^{s} given by X_{i} = \binom{p_{i}}{c_{i}}\binom{1}{1}^{ν_{i}−k_{i}}, where \binom{p_{i}}{c_{i}} is the matrix product M(L^{t^{i}})\binom{1}{ν_{i}}, k_{1} = m − 1 and k_{i} = 0 for all i ≥ 1. Then X_{i} ∈ T_{k_{i}}(L^{t^{i}}) for all i = 1, …, s.

5.39. Let τ = (m, T_{1}, T_{2}) ∈ 𝕋. For each (m, T_{1})-admissible sequence ν = (ν_{1}, …, ν_{s}), define τ_{ν} ∈ 𝕋 by τ_{ν} = (1, (T_{1})^{(ˇ s)}, X_{s} ⋯ X_{1}T_{2}), where (X_{1}, …, X_{s}) is determined by ν and (m, T_{1}) as in part 4 of 5.38. Then

[τ, ∞) = {τ} ∪ {τ_{ν} | ν is an (m, T_{1})-admissible sequence} .

Corollary 5.40. Let X be a surface satisfying (†) and suppose that (m, T′_{1}, T′_{2}) ∈ 𝕋(X). Let γ = γ(T′_{1}) and α(T′_{1}) = (α_{1}, α_{2}, …). Then the set

{(μ(T_{1}), μ(T_{2})) | (1, T_{1}, T_{2}) ∈ 𝕋(X) and (1, T_{1}, T_{2}) > (m, T′_{1}, T′_{2})}

is equal to

{ (μ(T′_{1}), μ(T′_{2}) · ∏_{i=1}^{s}(α_{i} + ν_{i}γ)) | (ν_{1}, …, ν_{s}) is (m, T′_{1})-admissible } .

5.41. We describe the elements of 𝕋_{0}(‡). Consider a triple τ = (m, T_{1}, T_{2}) where m is a positive integer and each T_{i} is a tableau (T_{i} ∈ T ) satisfying one of conditions (1–3) of 5.6 (each element of 𝕋_{0}(‡) is such a triple). Consider the connected component Γ of (G_{(−m)} ⊖ T_{1}) ⊖ T_{2} containing the vertices of G_{(−m)}. Then every connected component of (G_{(−m)} ⊖ T_{1}) ⊖ T_{2} is a linear chain and every vertex, except possibly the middle vertex of G_{(−m)} (which has weight −m), has weight strictly less than −1. So τ ∈ 𝕋_{0}(‡) ⇔ det(Γ) > 0 and in particular:

• If m > 1 then τ ∈ 𝕋_{0}(‡);

• if 1 ∈ {T_{1}, T_{2}} then τ ∈ 𝕋_{0}(‡).

Assume that m = 1 and that neither of T_{1}, T_{2} is empty; then T_{i} is either \binom{p_{i}}{c_{i}} or ( \begin{matrix}p_{i} & 1 \\ c_{i} & x_{i} \end{matrix}) with x_{i} ≥ 1. We may then compute det(Γ) in each case and conclude:

(1) If T_{1} = \binom{p_{1}}{c_{1}} and T_{2} = \binom{p_{2}}{c_{2}}, τ ∈ 𝕋_{0}(‡) ⟺ Δ > 0;

(2) if T_{i} = \binom{p_{i}}{c_{i}} and T_{j} = ( \begin{matrix}p_{j} & 1 \\ c_{j} & x_{j} \end{matrix} ), τ ∈ 𝕋_{0}(‡) ⟺ Δc_{j}x_{j} − c_{i} > 0;

(3) if T_{1} = ( \begin{matrix}p_{1} & 1 \\ c_{1} & x_{1} \end{matrix} ) and T_{2} = ( \begin{matrix}p_{2} & 1 \\ c_{2} & x_{2} \end{matrix} ), τ ∈ 𝕋_{0}(‡) ⟺ Δc_{1}c_{2}x_{1}x_{2} − c_{1}^{2}x_{1} − c_{2}^{2}x_{2} > 0,

where Δ = mc_{1}c_{2} − c_{1}p_{2} − c_{2}p_{1} = c_{1}c_{2} − c_{1}p_{2} − c_{2}p_{1}.

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Department of Mathematics and Statistics, University of Ottawa, Ottawa, Canada K1N 6N5

E-mail address: daniel@zenon.mathstat.uottawa.ca

Department of Mathematics and Statistics, McGill University, Montréal, Qc, Canada H3A 2K6

E-mail address: russell@math.mcgill.ca