ON LOG ℚ -HOMOLOGY PLANES AND WEIGHTED PROJECTIVE PLANES

DANIEL DAIGLE AND PETER RUSSELL

Let k be an algebraically closed field of characteristic zero and consider algebraic surfaces over k. Let ℳ be the class of normal affine surfaces U satisfying ML(U) = k, where ML(U) stands for the Makar-Limanov invariant of U, that is, the intersection of the kernels of all locally nilpotent derivations 𝒪_{U}(U) → 𝒪_{U}(U). The aim of this paper is to describe the class

ℳ_{0} = { U ∈ ℳ | Pic(U_{s}) is a finite group },

where U_{s} denotes U \ Sing(U). In the special case where k = ℂ, the class ℳ_{0} can also be defined as that of log ℚ-homology planes with trivial Makar-Limanov invariant. This explains our title but note that we will never assume that k = ℂ.

1. Statement of some results

Given positive integers a_{0}, a_{1}, a_{2}, the weighted projective plane ℙ(a_{0}, a_{1}, a_{2}) is defined by ℙ(a_{0}, a_{1}, a_{2}) = Proj R, where R = k[X_{0}, X_{1}, X_{2}] is the polynomial ring in three variables with the grading given by deg(X_{i}) = a_{i}. As is well-known, ℙ(a_{0}, a_{1}, a_{2}) is isomorphic to ℙ(a′_{0}, a′_{1}, a′_{2}) for some pairwise relatively prime positive integers a′_{0}, a′_{1}, a′_{2}. Whenever we consider a weighted projective plane ℙ(a_{0}, a_{1}, a_{2}), we will choose a_{0}, a_{1}, a_{2} to be pairwise relatively prime.

We use the set T_{∗}, defined in 3.1, to parametrize the class ℳ_{0}. One of our main results is: Theorem A. Given T ∈ T_{∗}, define (a_{0}, a_{1}, a_{2}) ∈ ℤ^{3} and f ∈ k[X_{0}, X_{1}, X_{2}] as follows:

(i) If T = \binom{p}{c} ∈ T_{1}, let (a_{0}, a_{1}, a_{2}) = (c − p, c, 1) and f = X_{1}.

(ii) If T = ( \begin{matrix}p & 1 \\ c & a \end{matrix}) ∈ T_{2}, there exists a_{2} ∈ ℤ satisfying:

(c − p)a_{2} ≡ 1 (mod c), gcd(a_{2}, a) = 1 and 0 < a_{2} < ac.

Choose any such a_{2} and define a_{0} = ac − a_{2}, a_{1} = a and f = X_{0}X_{2} + X_{1}^{c}. Then a_{0}, a_{1}, a_{2} are pairwise relatively prime positive integers and f is homogeneous with respect to the grading of k[X_{0}, X_{1}, X_{2}] given by deg X_{i} = a_{i} (i = 0, 1, 2). Let the surface U_{T} be the complement of the curve f = 0 in ℙ(a_{0}, a_{1}, a_{2}). Then:

(1) Up to isomorphism, U_{T} depends only on T (i.e., U_{T} is independent of the choice of a_{2} made in case (ii), in the above statement).

(2) U_{T} belongs to the class ℳ_{0} and each member of ℳ_{0} is a U_{T} for some T.

Research supported by CICMA and grants from NSERC Canada.

Research supported by grants from NSERC Canada.

(3) Given T, T′ ∈ T_{∗}, U_{T} is isomorphic to U_{T′} if and only if T′ ∈ {T, Ť} (see 3.3

for Ť). Remarks. (1) Since U_{T} is the complement of the curve f = 0 in ℙ(a_{0}, a_{1}, a_{2}), it

follows that it is a quotient of the hypersurface f = 1 of 𝔸^{3} by a finite cyclic group action.

More precisely, in case (ii) U_{T} is the quotient of the surface Y in 𝔸^{3} with equation X_{0}X_{2} + X_{1}^{c} = 1 by the group Ω_{ac} of ac-roots of 1, acting with weights −a_{2}, a = a_{1}, a_{2}. Since gcd(a_{2}, ac) = 1 we can pick j such that ja_{2} ≡ 1 (mod ac). Then j ≡ c − p (mod c) and we can replace the above weights by −1, a(c − p), 1. Precisely the points { (0, x_{1}, 0) | x_{1}^{c} = 1 } have non-trivial stabilizer, namely the group Ω_{a} of a-roots of 1. The invariants for Ω_{a} are generated by y_{0} = x_{0}^{a}, y_{2} = x_{2}^{a}, x_{0}x_{2}, y_{1} = x_{1}. Hence Y/Ω_{a}, which is the universal covering space of U_{T} in case k = ℂ, is the Danielewski surface with equation Y_{0}Y_{2} = (1 − Y_{1}^{c})^{a}.

In case (i) of Theorem A, a similar argument shows that U_{T} is a quotient of 𝔸^{2} by a cyclic group. So, if k = ℂ, U_{T} is contractible and is its own universal covering space.

(2) Some surfaces belonging to ℳ_{0} may be embedded in infinitely many (non isomorphic) weighted projective planes; one can see that this is the case for U_{T} if and only if T ∈ T_{1} (this particular claim is not proved in the present paper but see 7.2–7.5 for related questions). Also note that a_{2}, in Theorem A(ii), is not always unique.

Let us explain how Theorem A follows from the results of this paper. In 6.1 we define a map 𝔣̄: T_{∗} → \overline{ℳ}_{0}, where \overline{ℳ}_{0} is the set of isomorphism classes [U] of surfaces U ∈ ℳ_{0}. The fact that 𝔣̄ is a well-defined map is not trivial: It is based on 5.4, 3.9 and ultimately result 5.25 of [5]. In 7.1, we show that 𝔣̄(T) is the isomorphism class of the surface U_{T} defined in Theorem A; this proves assertion (1) of Theorem A, as well as the first half of assertion (2). The second half of (2) is the fact that 𝔣̄ is surjective (a consequence of 6.3) and assertion (3) is a rephrasing of 6.7.

See also result 6.9, which describes the properties of U_{T} (or 𝔣̄(T)) in terms of T. There is another way to present the classification of surfaces belonging to ℳ_{0}. Given

a normal surface U, let G_{res}[U] denote the resolution graph of U (see 2.1) and let G_{∞}[U] denote the equivalence class of weighted graphs which contains the dual graph of any divisor at infinity of U (see the beginning of section 2 for details). Then 6.9 contains a description of the set

(1)  { (G_{∞}[U], G_{res}[U]) | U ∈ ℳ_{0} }.

LOG ℚ -HOMOLOGY PLANES AND WEIGHTED PROJECTIVE PLANES 3

From this description, it follows in particular that { G_{res}[U] | U ∈ ℳ_{0} } is exactly the set of all admissible chains (an admissible chain is a weighted graph which is a linear chain and in which every weight is strictly less than −1; the empty graph is considered to be an admissible chain). It also follows that an equivalence class 𝒞 of weighted graphs is an element of { G_{∞}[U] | U ∈ ℳ_{0} } if and only if some element of 𝒞 has the form:

(2)  [figure: …; 0; −1; ω_{1}; ω_{q}]  (where q ≥ 0 and ∀_{i} ω_{i} ≤ −2).

To decide whether two graphs of the form (2) are equivalent, see 2.17. We have: Theorem B (See 6.5). Let U and U′ be surfaces belonging to the class ℳ_{0}. If G_{∞}[U] = G_{∞}[U′] and G_{res}[U] = G_{res}[U′] then U ≅ U′.

Clearly, this result and the description (6.9) of the set (1) constitute a classification of ℳ_{0}. Note that it is easy to go back and forth between T ∈ T_{∗} and the graphs G_{∞}[U_{T}] and G_{res}[U_{T}], see section 8.

We also classify G_{a}-actions on these surfaces. Given a surface U = Spec R belonging to ℳ, define:

klnd(R) = { ker D | D : R → R is a nonzero locally nilpotent derivation },

which is a collection of subalgebras of R. It is well-known that the problem of describing the G_{a}-actions on U is equivalent to that of describing the set klnd(R). One can show that klnd(R) has the same cardinality as the ground field k (whenever U ∈ ℳ), but it is more interesting to count the orbits, relative to the obvious action of the group Aut_{k}(R) on the set klnd(R) (that is, if θ ∈ Aut_{k}(R) and A ∈ klnd(R), define θA = θ(A)). For instance, it was suggested that the action might be transitive whenever U ∈ ℳ is a smooth surface. The following fact shows that that idea was not correct: Theorem C. Let T ∈ T_{∗} and let U_{T} = Spec R be the surface defined in Theorem A. Then the action of Aut_{k}(R) on klnd(R) is such that:

(1) The number of orbits is either one or two.

(2) The action is transitive if and only if T = Ť.

This result is a reformulation of 6.8, taking into account the fact (7.1) that 𝔣̄(T) is the isomorphism class of U_{T}.

We may give a geometric interpretation of the condition T = Ť. Let us say that a surface U ∈ ℳ is symmetric at infinity if the following equivalent conditions are satisfied:

• For some element of G_{∞}[U] of the form (2), (ω_{1}, …, ω_{q}) = (ω_{q}, …, ω_{1}).

• For every element of G_{∞}[U] of the form (2), (ω_{1}, …, ω_{q}) = (ω_{q}, …, ω_{1}).

(Equivalence follows from 2.17.) Now, by 6.9, the condition T = Ť is equivalent to U_{T} being symmetric at infinity; so Theorem C implies:

The action is transitive ⟺ U_{T} is symmetric at infinity

where in fact “ ⟹ ” is valid for all U ∈ ℳ.

In the case k = ℂ, some of our results can also be found in [10] (see also [11] for the smooth case). That paper describes the set (1) and shows that each member of ℳ_{0} can be realized as a quotient of the type described in the remark following Theorem A.

However, [10] does not give embeddings into weighted projective planes (i.e., Theorem A(2)) and does not address the question of deciding when two members of ℳ_{0} are isomorphic (i.e., Theorem A(3) or Theorem B). Also, Theorem C is not contained in [10]. Note that [10] investigates the relation between the condition ML(U) = ℂ and finiteness of the fundamental group at infinity of U; we don’t consider that question here.

This work is, in essence, an elaborate corollary of our results in [5] and [6] on the classification of affine rulings of weighted projective planes. Theorem A(3) and Theorem C, in particular, are quite delicate results that appear to require the very precise methods developed there.

2. Preliminaries on surfaces

Let k be an algebraically closed field of characteristic zero. All algebraic varieties are assumed to be k-varieties. All divisors are Weil divisors.

Let D be a divisor of a complete normal surface S. By a component of D we always mean an irreducible component. We call D an SNC-divisor of S if it satisfies:

(a) D is effective and reduced

(b) supp(D) ⊂ S \ Sing(S)

(c) each component of D is a smooth curve

(d) if C, C′ are distinct components of D then C · C′ ≤ 1

(e) if C, C′, C′′ are distinct components of D then C ∩ C′ ∩ C′′ = ∅.

If D is an SNC-divisor of S then we may consider the dual graph of D in S, which we denote G(D, S). For the definition of G(D, S) and the theory of weighted graphs, see for instance [13].

By “graph” we mean a finite undirected graph in which no edge connects a vertex to itself and at most one edge joins any given pair of vertices. A vertex adjacent to a vertex v is also called a neighbor of v; we say that v is a branch point if it has more than two neighbors. A tree without branch points is called a linear chain. An admissible chain is a (possibly empty) weighted graph which is a linear chain and in which every weight is strictly less than −1. If D is an SNC-divisor of S then a branching component of D is an irreducible component which is a branch point in the dual graph G(D, S).

If U is a normal surface then there exists a smooth-normal compactification of U, by which we mean an open immersion u : U ↪ S such that S is a complete normal surface and S \ U is the support of an SNC-divisor of S (so in particular Sing(S) = Sing(U)). If u : U ↪ S is a smooth-normal compactification of U then we define G(u) = G(D, S), where D is the SNC-divisor of S satisfying S \U = supp(D). Note that the equivalence class of the weighted graph G(u) depends only on the isomorphism class [U] of U. This equivalence class of weighted graphs is denoted G_{∞}[U].

A normal surface U is said to be completable by rational curves if there exists a smooth-normal compactification u : U ↪ S of U such that every curve in S \ U is a rational curve. If this is the case then every smooth-normal compactification of U has that property.

If U is a normal surface then

{ G(u) | u is a smooth-normal compactification of U } is a nonempty subset of G_{∞}[U]; it is equal to G_{∞}[U] whenever U is completable by rational curves.

If f : X → Y is a birational morphism of surfaces, we use the notations

cent(f) = { P ∈ Y | f^{−1}(P) contains more than one point }
exc(f) = f^{−1}( cent(f))

for the center and exceptional locus of f respectively. 2.1. Recall from [6]-1.19 that the resolution graph of a normal surface U is the dual graph of ℰ in Û, where ℰ is the exceptional locus of the minimal SNC resolution of singularities Û → U. We will denote that weighted graph by G_{res}[U], which reminds us that the resolution graph depends only on the isomorphism class [U].

G_{a}-actions and 𝔸^{1}-fibrations

2.2. Notations. In addition to the classes ℳ and ℳ_{0} defined in the introduction, we will also consider the larger class 𝒩 of normal algebraic surfaces.

Given a class 𝒮 of algebraic surfaces (e.g., ℳ, ℳ_{0} or 𝒩), let

𝒮^{+} = { (U, ρ) | U ∈ 𝒮 and ρ : U → 𝔸^{1} is a morphism whose general fiber is 𝔸^{1}}.

Note that surjectivity of the map ρ : U → 𝔸^{1} is not required in the definition of 𝒮^{+}. Define an equivalence relation on the set 𝒮^{+} by declaring that (U, ρ) ∼ (U′, ρ′) if there exists a commutative diagram:

U ⟶ U′
ρ↓  ↓^{ρ′}
𝔸^{1} ⟶ 𝔸^{1}

where the horizontal arrows are isomorphisms of varieties. Let [U, ρ] denote the equivalence class of the pair (U, ρ) and [U] the isomorphism class of U. Then define:

\overline{𝒮} = \{ [U] | U ∈ 𝒮 \} and \overline{𝒮}^{+} = \{ [U, ρ] | (U, ρ) ∈ 𝒮^{+} \}.

We also consider the set map \overline{𝒮}^{+} ⟶ \overline{𝒮}, defined by [U, ρ] ↦ [U], which we simply call the projection. By 2.3, the projection is surjective when 𝒮 = ℳ or ℳ_{0}. 2.3. Let U = Spec R be a normal affine surface and recall, from the introduction, that the study of G_{a}-actions on U reduces to describing the set klnd(R). The following facts are well-known and easy to prove.

(1) ML(U) = k if and only if klnd(R) contains more than one element.

(2) If ML(U) = k then R^{∗} = k^{∗} and U is rational and completable by rational curves.

(3) If ρ : U → 𝔸^{1} is a surjective morphism with general fiber 𝔸^{1}, then the image of the corresponding homomorphism ρ^{∗} : k[t] ↪ R is an element of klnd(R). If U is rational and R^{∗} = k^{∗} then all elements of klnd(R) can be obtained in this way.

(4) Two morphisms U → 𝔸^{1} as in statement (3) correspond to the same element

of klnd(R) if and only if they differ by an automorphism of 𝔸^{1}. 2.4. This paragraph makes the link between Theorem C and the other results of this paper. Consider a surface U = Spec(R) belonging to ℳ. Let E ⊂ \overline{ℳ}^{+} be the inverse image of [U] ∈ \overline{ℳ} by the projection \overline{ℳ}^{+} → \overline{ℳ}.

Fix A ∈ klnd(R); by 2.3, there exist surjective morphisms ρ : U → 𝔸^{1} with general fiber 𝔸^{1} and satisfying A = im(ρ^{∗}). Moreover, any two such morphisms differ by an automorphism of 𝔸^{1} and consequently the element [U, ρ] of E is completely determined by A. This defines a set map

\begin{matrix}F : klnd(R) & ⟶ & E \\ A & ⟼ & [U, ρ] \end{matrix}

Note that F is surjective (R^{∗} = k^{∗} by 2.3, so every dominant morphism ρ : U → 𝔸^{1} is surjective) and that, given A, A′ ∈ klnd(R),

F (A) = F (A′) ⟺ ∃_{θ∈Aut_{k}(R)} θ(A) = A′.

Thus, if we consider the natural action of the group Aut_{k}(R) on the set klnd(R), the number of orbits is equal to the cardinality of E.

Completions and minimal completions

2.5. Definition. Let (U, ρ) be an element of 𝒩^{+} (see 2.2). A completion of (U, ρ) is a commutative diagram

(3)  \begin{matrix}U & ⟶^{u} & S \\ ^{ρ}↓ & & ↓ρ̄ \\ 𝔸^{1} & ⟶ & ℙ^{1} \end{matrix}

where u : U → S is a smooth-normal compactification of U, 𝔸^{1} → ℙ^{1} is an open immersion and ρ̄ is a morphism.

Let D be the SNC-divisor of S such that S \ U = supp(D). A component C of D is called ρ-vertical if ρ̄(C) is a point; if C is not ρ-vertical, we call it ρ-horizontal. The assumption that the general fiber of ρ is 𝔸^{1} implies that the general fiber of ρ̄ meets D in one point. It easily follows that

D has exactly one ρ-horizontal component

and that, if the unique ρ-horizontal component is denoted H, then

ρ̄ restricts to an isomorphism H → ℙ^{1}.

We call D and H the boundary divisor and the horizontal component of the completion (3), respectively.

It is clear that each element (U, ρ) of 𝒩^{+} has a completion. Observe that given any completion (3) of (U, ρ), the morphism ρ̄ : S → ℙ^{1} is a ℙ^{1}-fibration (i.e., the general fiber of ρ̄ is a ℙ^{1}). ℙ^{1}-fibrations have been studied extensively, and the structure of their fibers is understood. Since all ρ-vertical components of the boundary divisor D are (by definition) contained in fibers of ρ̄, one can obtain information about the structure of D. In particular, the properties of ℙ^{1}-fibrations imply the existence of minimal completions: 2.6. Definition. Let (U, ρ) be an element of 𝒩^{+}. By a minimal completion of (U, ρ), we mean a completion (3) of (U, ρ) satisfying the additional condition:

Every ρ-vertical component C of the boundary divisor D satisfies C^{2} ≠ −1. The following is well-known:

2.7. Lemma. Given any (U, ρ) ∈ 𝒩^{+} and x ∈ ℤ, there exists a minimal completion of (U, ρ) whose horizontal component H satisfies H^{2} = x.

We shall now see that each element [U, ρ] of \overline{𝒩}^{+} (see 2.2) determines a specific element G_{∞}[U, ρ] of G_{∞}[U]. 2.8. Definition. Let (U, ρ) be an element of 𝒩^{+}. By 2.7, we may choose a minimal completion (3) of (U, ρ) whose horizontal component H satisfies H^{2} = −1. Let D be the boundary divisor of such a completion. We claim:

The weighted graph G(D, S) is independent of the choice of the completion. In fact, it depends only on [U, ρ].

Assuming that this is true, define G_{∞}[U, ρ] = G(D, S) and note that G_{∞}[U, ρ] ∈ G_{∞}[U].

The meaning of the above claim is that [U, ρ] ⟼ G_{∞}[U, ρ]

is a well-defined map, going from \overline{𝒩}^{+} to the set of all weighted graphs. This claim is justified by part (2) of 2.15, below.

Note that G_{∞}[U, ρ] is not independent of ρ.^{1} For instance, there exists U ∈ ℳ_{0} and morphisms ρ, ρ′ : U → 𝔸^{1} (whose general fibers are 𝔸^{1}) such that G_{∞}[U, ρ] and G_{∞}[U, ρ′] are (79) and (81) respectively. The proof of Theorem C is based on this type of phenomenon.

To prove that Definition 2.8 is sound, as well as several other results of this paper, the following combinatorial object is very convenient: 2.9. Definition. A weighted pair is an ordered pair 𝒫 = (G, v) where G is a nonempty weighted graph and v is a vertex of G. We call v and G the distinguished vertex and the underlying weighted graph of 𝒫. One defines an equivalence relation ≈ on the set of weighted pairs by declaring that (G, v) ≈ (G′, v′) means that G can be transformed into G′ by a finite sequence of blowings-up and blowings-down of weighted graphs, in such a way that the distinguished vertex is preserved. Refer to [6]-2.5 for the precise definition of ≈. Note in particular that the condition (G, v) ≈ (G′, v′) implies that G and G′ are equivalent weighted graphs.

1So result 1.9 of [2] (as far as we understand it) is false.

2.10. Definition. Let (U, ρ) be an element of 𝒩^{+}. Given a completion (3) of (U, ρ), with boundary divisor D and horizontal component H, define the weighted pair

𝒫(U, ρ, u) = (G(D, S), H).

We will see in 2.15 that, up to equivalence ≈ of weighted pairs, 𝒫(U, ρ, u) is independent of the choice of a completion. 2.11. Definition. We say that a weighted pair 𝒫 = (G, v) is of type (⋆) if it satisfies the conditions:

(1) Every connected component of G is a tree.

(2) If w is the weight of a vertex of G \ {v} then w = 0 or w ≤ −2.

(3) Some vertex of G \ {v} has weight zero. Moreover, any such vertex has exactly

one neighbor in G and that neighbor is v. We say that 𝒫 is of type (⋆⋆) if it satisfies (1–3) and:

(4) G is connected and exactly one vertex of G \ {v} has weight zero.

From the properties of ℙ^{1}-fibrations, we now derive: 2.12. Lemma. If (3) is a minimal completion of (U, ρ) ∈ 𝒩^{+} then 𝒫(U, ρ, u) is a weighted pair of type (⋆). If we also assume that U is affine and has trivial units (i.e., 𝒪_{U}(U)^{∗} = k^{∗}), then 𝒫(U, ρ, u) is of type (⋆⋆).

Proof. Let the boundary divisor and horizontal component of the minimal completion (3) be denoted by D and H respectively. Let G = G(D, S), then 𝒫(U, ρ, u) = (G, H).

It is known that every fiber of ρ̄ is a tree; since distinct fibers are disjoint, and since D has only one horizontal component, it follows that 2.11(1) holds.

Let C be a vertex of G \ {H}. Then C is a component of a fiber of ρ̄. From known properties of the fibers, it follows that C^{2} ≤ 0. Since C is a ρ-vertical component of D, and since (3) is a minimal completion, C^{2} ≠ −1 and 2.11(2) holds.

We may pick a point P of ℙ^{1} which is not in the image of the composite U ⟶^{ρ} 𝔸^{1} ↪ ℙ^{1}. Consider the fiber L = ρ̄^{−1}(P). Then L is entirely contained in supp(D) and, consequently, no component of L has self-intersection equal to −1. From known properties of the fibers of ρ̄, it follows that L is an irreducible curve and that L^{2} = 0. Hence L is a vertex of G \ {H} and has weight zero.

Let L be any vertex of weight zero in G \ {H}. Then L is an irreducible component of some fiber ρ̄^{−1}(P), and L^{2} = 0. From known properties of the fibers of ρ̄, it follows that L = ρ̄^{−1}(P). Thus L ∩ H ≠ ∅ and (since distinct fibers are disjoint) H is the only neighbor of L. This proves 2.11(3) and hence that 𝒫(U, ρ, u) is of type (⋆).

Note that the above argument shows that the number of vertices of weight zero in G \ {H} is equal to the cardinality of ℙ^{1} minus image of U ⟶^{ρ} 𝔸^{1} ↪ ℙ^{1}.

As is well-known, if U is affine then G is connected; if also U has trivial units then every dominant morphism U → 𝔸^{1} is surjective, so ρ : U → 𝔸^{1} is surjective. By the above paragraph, there is only one vertex of weight zero in G \ {H}, so 𝒫(U, ρ, u) is of type (⋆⋆). □

2.13. Definition. A weighted pair 𝒫 = (G, v) is minimal if every vertex in G \ {v} of weight −1 has more than two neighbors in G.

Note that every weighted pair of type (⋆) is minimal. We leave it to the reader to verify: 2.14. Lemma. Consider equivalent weighted pairs 𝒫 ≈ 𝒫′, both of which are minimal. If one of them is of type (⋆) then 𝒫 and 𝒫′ are identical, except possibly for the weight of the distinguished vertex. In particular, both of them are of type (⋆). 2.15. Lemma. Let (U, ρ) and (U′, ρ′) be equivalent elements of 𝒩^{+} and let

\begin{matrix}U & ⟶^{u} & S \\ ^{ρ}↓ & & ↓ρ̄ \\ 𝔸^{1} & ⟶ & ℙ^{1} \end{matrix} and \begin{matrix}U′ & ⟶^{u′} & S′ \\ ^{ρ′}↓ & & ↓^{}^{ρ̄′} \\ 𝔸^{1} & ⟶ & ℙ^{1} \end{matrix}

be completions of (U, ρ) and (U′, ρ′) respectively.

(1) 𝒫(U, ρ, u) ≈ 𝒫(U′, ρ′, u′)

(2) If both completions are minimal then the weighted pairs 𝒫(U, ρ, u) and 𝒫(U′, ρ′, u′) are identical, except possibly for the weight of the distinguished vertex.

Proof. Let D be the SNC-divisor of S such that U = S \ supp(D) and let H be the unique ρ-horizontal component of D; let D′ and H′ be the corresponding objects in the second diagram. There exists a birational isomorphism β : S −−→ S′ which restricts to an isomorphism U → U′ and which gives a commutative diagram:

(4)  \begin{matrix}S & −−→^{β} & S′ \\ ρ̄ ↓ & & ↓ ρ̄′ \\ ℙ^{1} & ⟶^{≅} & ℙ^{1} \end{matrix}

We may consider a complete normal surface W and birational morphisms S ⟵^{w} W ⟶^{w′} S′ such that w′ ∘ w^{−1} = β. Let H̃ ⊂ W be the proper transform of H ⊂ S with respect to w. Since every fiber of ρ̄ meets H, it follows that every fiber of ρ̄ ∘ w meets H̃; so, by commutativity of (4), every fiber of ρ̄′ ∘ w′ meets H̃; thus w′ cannot contract H̃ to a point, i.e., w′(H̃) ⊂ S′ is one of the components of D′. Since H′ is the only component of D′ which is ρ′-horizontal, we must have

w′(H̃) = H′.

Hence, by blowing-up and blowing-down, it is possible to transform G(D, S) into G(D′, S′) without ever contracting H. So, by definition [6]-2.5 of ≈, we have

𝒫(U, ρ, u) = (G(D, S), H) ≈ (G(D′, S′), H′) = 𝒫(U′, ρ′, u′),

which proves assertion (1).

By 2.14, if 𝒫 and 𝒫′ are weighted pairs of type (⋆) and satisfying 𝒫 ≈ 𝒫′, then 𝒫 and 𝒫′ are identical, except possibly for the weight of the distinguished vertex. So assertion (2) follows from assertion (1) together with 2.12. □

Remark. By part (2) of 2.15, the claim contained in 2.8 is true.

Some graph theory

2.16. Definition. A weighted graph G is called a chain of type (Z) if it is a linear chain of the form:

(5)  G = [figure: …; 0; x; ω_{1}; ω_{q}]

where q ≥ 0, x is any integer and ω_{1}, …, ω_{q} ∈ ℤ are such that ω_{i} ≤ −2 for all i. We define the bideterminant of G to be the ordered pair

bidet G = ( det ([figure: …; ω_{1}; ω_{2}; ω_{q}]), det ([figure: …; ω_{2}; ω_{q}])) ∈ ℕ^{2},

where det([figure: …; ω_{1}; ω_{2}; ω_{q}]) and det([figure: …; ω_{2}; ω_{q}]) are determinants of weighted graphs (see [6]-1.10), and where we adopt the convention that bidet G = (1, 0) if q = 0, and that bidet G = (−ω_{1}, 1) if q = 1. Since the ordered q-tuple (ω_{1}, …, ω_{q}) ∈ ℤ^{q} is uniquely determined by G, so is bidet G. So every chain of type (Z) has a bideterminant. 2.17. Lemma. The set of chains of type (Z) equivalent to the chain (5) is:

{ [figure: …; 0; y; ω_{1}; ω_{q}] | y ∈ ℤ } ∪ { [figure: …; 0; y; ω_{q}; ω_{1}] | y ∈ ℤ } .

Proof. Immediate consequence of the classification given in [3]. Alternatively, but this requires some work, the assertion may be derived from section 3 of [5], which gives several results on chains of type (Z). □

The next result will be used in the proof of 2.19. If G, H are weighted graphs, we write G ← H or H → G to indicate that H is obtained by blowing-up G (or equivalently that G is obtained by blowing-down H). The arrow should not be interpreted as a map, and the direction of the arrow reminds us of the geometric situation. Writing G ⟵^{π} H allows us to speak of “the blowing-up π”. We say that π is subdivisional if it is the blowing-up of G at an edge. 2.18. Lemma. Let G_{0} ← G_{1} ← ⋯ ← G_{n} (where n ≥ 1) be a sequence of blowings-up of weighted graphs and, for i ∈ {1, …, n}, let e_{i} be the vertex of G_{i} created by the blowing-up G_{i−1} ← G_{i}. Assume:

(i) G_{0} is the underlying weighted graph of a weighted pair of type (⋆⋆).

(ii) (G_{n}, e_{n}) is equivalent to a weighted pair of type (⋆⋆). Then G_{0} is a chain of type (Z).

Proof. Given a weighted graph G, the notation x ∈ G means that x is a vertex of G, w(x, G) denotes the weight of the vertex x in G and the symbol G ≪ 0 means that for every x ∈ G we have w(x, G) < −1.

Suppose that the result is false and consider a counterexample

(6)  G_{0} ⟵^{π_{1}} G_{1} ⟵^{π_{2}} ⋯ ⟵^{π_{n}} G_{n}

which minimizes n. Starting at G_{n}, perform a sequence of blowings-down

(7)  G_{n} = H_{0} ⟶^{ν_{1}} H_{1} ⟶^{ν_{2}} ⋯ ⟶^{ν_{m}} H_{m}

in such a way that e_{n} is not contracted (so e_{n} is a vertex of H_{m}) and (H_{m}, e_{n}) is a minimal weighted pair. Then (H_{m}, e_{n}) ≈ (G_{n}, e_{n}), so assumption (ii) and 2.14 imply that

(8)  (H_{m}, e_{n}) is of type (⋆⋆).

By (i), there exists a vertex v of G_{0} such that

(9)  (G_{0}, v) is of type (⋆⋆).

This and the fact that G_{0} is not a chain of type (Z) imply that G_{0} is not equivalent to a linear chain. Thus:

(10)  No G_{i} or H_{j} is equivalent to a linear chain.

Recall that, if 1 ≤ i < j ≤ n, the vertex set of G_{i} is naturally embedded in that of G_{j}. In particular, e_{1}, …, e_{n} may be viewed as vertices of G_{n}. We begin by showing:

(11)  { i | w(e_{i}, G_{n}) = −1 } = {n}

or equivalently:

(12)

For all i such that 1 < i ≤ n, G_{i} is a blowing-up of G_{i−1} at e_{i−1} or at an edge

incident to e_{i−1}.

If these conditions do not hold then we may change the order of the blowings-up in (6) in such a way that the vertex e_{n} is created at an earlier stage of the blowing-up process; more precisely, there exists a sequence

(13)  G_{0} = G′_{0} ⟵^{π′_{1}} G′_{1} ⟵^{π′_{2}} ⋯ ⟵^{π′_{n}} G′_{n}

satisfying the following condition: Let e′_{i} denote the vertex of G′_{i} created by G′_{i−1} ← G′_{i}; then there exists an isomorphism of weighted graphs θ : G_{n} → G′_{n} such that θ(e_{n}) = e′_{p}, where 1 ≤ p < n. Identifying G_{n} with G′_{n}, we may write:

(G′_{p}, e′_{p}) ≈ (G′_{n}, e′_{p}) = (G_{n}, e_{n}) ≈ some weighted pair of type (⋆⋆).

This means that G′_{0} ← ⋯ ← G′_{p} is a counterexample, contradicting minimality of (6). This proves that (11) and (12) hold.

Since e_{n} is created by a blowing-up, G_{n} has at most two branches at e_{n}; in fact, it has exactly two. Indeed, (8) implies that some branch ℬ of G_{n} at e_{n} satisfies:

(14)  ℬ ∪ {e_{n}} can be contracted to [figure: 0; e_{n}]

and if there is no other branch then we get a contradiction with (10). Let ℬ^{∗} be the other branch of G_{n} at e_{n}. To avoid a contradiction with (10), we must have:

(15)  ℬ^{∗} ∪ {e_{n}} is not a linear chain.

We claim:

(16)  π_{1}, …, π_{n} are subdivisional.

We already know that π_{n} is subdivisional, because G_{n} has two branches at e_{n}. So, in view of (12), if (16) is false then some branch ℬ′ of G_{n} at e_{n} must satisfy:

ℬ′ ∪ {e_{n}} is a linear chain and ℬ′ ≪ 0.

Then ℬ′ ≠ ℬ by (14) and ℬ′ ≠ ℬ^{∗} by (15), which is absurd. This proves (16).

Let z be the unique vertex of weight zero in G_{0} \ {v}; let 𝒜 and 𝒜′ be the two branches of G_{n} at e_{n}, where z ∈ 𝒜 (thus {𝒜, 𝒜′} = {ℬ, ℬ^{∗}}).

If w(z, G_{n}) = 0 then z and v are neighbors in G_{n}, so z, v ∈ 𝒜. Then (11) gives

{ x ∈ G_{n} | w(x, G_{n}) ≥ −1 } ⊆ {z, v, e_{n}} ⊆ 𝒜 ∪ {e_{n}},

so 𝒜′ ≪ 0, so 𝒜′ ≠ ℬ by (14), so ℬ = 𝒜 and z, v ∈ ℬ. Then ℬ contains at least two vertices and one of them has weight 0; this makes (14) impossible.

This proves that w(z, G_{n}) ≠ 0, so π_{1} is the blowing-up of G_{0} at the edge {z, v}. Then (16) implies that 𝒜 ∪ {e_{n}} is a linear chain, so 𝒜 ≠ ℬ^{∗} by (15). So ℬ = 𝒜 and z ∈ ℬ.

To summarize, conditions (12) and (16), together with

z ∈ ℬ and π_{1} is the blowing-up of G_{0} at the edge {z, v},

imply that ℬ ∪ {e_{n}} has the following form:

[figure: …; x_{1}; z; x_{2}; x_{q}; −1; e_{n}]

where q ≥ 1, x_{1} ≤ −1 and x_{j} ≤ −2 for all j ≥ 2. This makes (14) impossible, so we are done. □

A characterization of elements of ℳ

It is known that, for a normal affine surface U, the implication

(17)  ML(U) = k ⟹ G_{∞}[U] contains a linear chain

is true (smooth case: [2] or [1], which are based on earlier work of Gizatullin: [8], [9]; general case: [10]). However the converse is false, as shown by the example U = 𝔸_{∗}^{1}×𝔸^{1} (where 𝔸_{∗}^{1} is 𝔸^{1} minus a point).

The next two results are improved versions of (17). Because of the confusion associated with this type of result ((17) is sometimes stated and proved as an “if and only if” statement, which is incorrect) and because all available proofs assume either that k = ℂ or that U is smooth, we decided to include the proofs of 2.19 and 2.20. Note that (17) is a consequence of 2.19. 2.19. Proposition. Let (U, ρ) ∈ ℳ^{+}. If (3) is any minimal completion of (U, ρ), and if D denotes its boundary divisor, then G(D, S) is a chain of type (Z).

Proof. Consider the weighted pair 𝒫(U, ρ, u) = (G(D, S), H), where D and H are the boundary divisor and the horizontal component of the minimal completion (3) being considered. Let B = 𝒪_{U}(U), then 2.3 gives B^{∗} = k^{∗} and 2.12 implies that 𝒫(U, ρ, u) is of type (⋆⋆).

Let Z be the unique component of D satisfying Z ≠ H and Z^{2} = 0. The assumption ML(U) = k implies that U is rational, so the complete linear system |Z| on S is one-dimensional, base point free and has general member isomorphic to ℙ^{1}. The corresponding morphism S → ℙ^{1} is the ρ̄ of (3).

In view of 2.3, the assumption ML(U) = k implies that there exists a morphism ρ′ : U → 𝔸^{1} whose general fiber is an 𝔸^{1} and which is “genuinely different” from ρ, in the following sense: Extend ρ′ to a rational map S −−→ ℙ^{1} and consider the corresponding linear system Λ on S; then Λ ≠ |Z|. Consider the minimal resolution of the base points^{2} of Λ (where n = 0 if Bs(Λ) = ∅):

(18)  S_{n} ⟶^{π_{n}} ⋯ ⟶^{π_{1}} S_{0} = S

where each π_{i} is the blowing-up of S_{i−1} at a smooth point P_{i}. Write E_{i} = π_{i}^{−1}(P_{i}) ⊂ S_{i} and let u_{n} : U ↪ S_{n} be the open immersion obtained by factoring u through (18). Now ρ′ extends to a morphism ρ̄′ : S_{n} → ℙ^{1} and

(19)  \begin{matrix}U & ⟶^{u_{n}} & S_{n} \\ ^{ρ′}↓ & & ↓^{}^{ρ̄′} \\ 𝔸^{1} & ⟶ & ℙ^{1} \end{matrix}

is a completion of (U, ρ′).

Consider the case where Bs(Λ) = ∅; then n = 0, S_{n} = S and u_{n} = u, so (19) is a completion of (U, ρ′) with boundary divisor D. If Z is ρ′-vertical then the fact that Z^{2} = 0 implies that Z is equal to a fiber of ρ̄′; then Z ∈ Λ, so Λ ⊆ |Z| (because |Z| is a complete linear system), so Λ = |Z| (because dim |Z| = 1 = dim Λ), which is absurd. Hence, Z must be ρ′-horizontal, so every component of D − Z is ρ′-vertical. Since Z has only one neighbor in G(D, S) (namely, H), supp(D − Z) is connected and consequently is included in a fiber of ρ̄′; since some fiber of ρ̄′ is entirely contained in D, supp(D −Z) is equal to a fiber. By properties of ℙ^{1}-fibrations, if the fiber supp(D −Z) is not irreducible then it contains a (−1)-curve other than H; this is not the case by minimality of (3), so D − Z = H and G(D, S) is a chain of type (Z).

From now-on, suppose that Bs(Λ) ≠ ∅ and consider the weighted pair 𝒫(U, ρ′, u_{n}) associated to the completion (19). Since n ≥ 1 and (18) is minimal, E_{n} is the horizontal component of (19); so E_{n} is the distinguished vertex of 𝒫(U, ρ′, u_{n}). We have 𝒫(U, ρ′, u_{n}) ≈ 𝒫′ by 2.15, where 𝒫′ is the weighted pair associated to a minimal completion of (U, ρ′). Since 𝒫′ is of type (⋆⋆) by the first paragraph of the proof, the desired conclusion follows from 2.18. □

2.20. Theorem. Let U be a normal affine surface. Then ML(U) = k if and only if the following conditions are satisfied:

(a) U is rational and completable by rational curves

(b) G_{∞}[U] contains a chain of type (Z).

Proof. It is well-known that ML(U) = k implies condition (a), and 2.19 implies (b).

2Actually, Λ has at most one base point on S, but we don’t need to know this for the proof.

Conversely, let U be a normal affine surface satisfying conditions (a) and (b) and let B = 𝒪_{U}(U). We shall show that klnd(B) is an infinite set, which will prove the “if” part of 2.20.

We may choose a smooth-normal compactification u : U ↪ S of U such that G(u) is a chain of type (Z). Let D be the SNC-divisor of S such that S \ U = supp(D) and consider the components Z and Σ of D such that G(u) = G(D, S) is the following:

(20)  [figure: …; 0; Z; −1; Σ; ω_{1}; ω_{q}]

where q ≥ 0 and ω_{1}, …, ω_{q} ≤ −2 (by 2.17, it is indeed possible to arrange Σ^{2} = −1; we choose a smooth-normal compactification u which has this property).

Let P ∈ Z \ Σ. We now proceed to construct a surjective morphism ρ : U → 𝔸^{1} with general fiber 𝔸^{1} and which satisfies:

(21)

If we extend ρ to a rational map S −−→ ℙ^{1} and consider the corresponding linear system Λ on S, then Bs(Λ) = {P }.

In view of 2.3, this will imply that klnd(B) is an infinite set.

By 2.17, the chains (20) and

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

are equivalent weighted graphs. In fact, Theorem 3.32 of [5] is more precise: It describes a sequence of blowings-up and blowings-down which transforms (20) into (22). From that description, and since we assumed that U is completable by rational curves (and hence all components of D are rational), it follows that there exists birational morphisms

S ⟵^{π} S^{∗} ⟶^{π′} S′

of projective normal surfaces satisfying the following conditions:

(i) cent(π) = {P } and cent(π′) is a smooth point of S′.

(ii) π^{−1}(supp(D)) is the support of an SNC-divisor D^{∗} of S^{∗} and supp(D^{∗}) =

exc(π) ∪ exc(π′). (iii) π′(supp(D^{∗})) is the support of an SNC-divisor D′ of S′ such that G(D′, S′) is

the chain (22). Let U′ be the complement of supp(D′) in S′, let Z′ be the unique component of D′

of self-intersection 0 and let Σ′ be the unique component of D′ satisfying Σ′ ≠ Z′ and Σ′ ∩ Z′ ≠ ∅ (refer to (22)). Since U (and hence S′) is rational by assumption, it is well-known that the complete linear system |Z′| on S′ is one-dimensional, base point free, and has general member isomorphic to ℙ^{1}; hence |Z′| gives rise to a ℙ^{1}-fibration f̄: S′ → ℙ^{1}. Consider the point P_{∞} ∈ ℙ^{1} defined by f̄(Z′) = {P_{∞}}. Since the fiber f̄^{−1}(P_{∞}) = Z′ is disjoint from U′, we have f̄(U′) ⊆ ℙ^{1} \ {P_{∞}}, so f̄ restricts to a morphism f : U′ → 𝔸^{1} whose general fiber is an 𝔸^{1}. By properties of ℙ^{1}-fibrations, the fact that ω_{1}, …, ω_{q} ≤ −2 implies that Z′ is the only fiber of f̄ which is entirely contained in supp(D′). Thus f : U′ → 𝔸^{1} is surjective. Observe that π′ ∘ π^{−1} restricts to an isomorphism θ : U → U′ and define ρ = f ∘ θ : U → 𝔸^{1}. Then ρ is surjective and has general fiber 𝔸^{1}.

Define Λ as in (21) and note that Λ corresponds to |Z′| via π′ ∘ π^{−1}. Note that Σ′ is a section of |Z′| and let Σ̃′ ⊂ S^{∗} denote the strict transform of Σ′. Since supp(D^{∗}) = exc(π) ∪ exc(π′), and since Σ̃′ is obviously not shrunk by π′, we have Σ̃′ ⊆ exc(π). So π(Σ̃′) = {P }, which proves (21) and the “if” part of 2.20. □

3. Tableaux and affine rulings

3.1. Definition. 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 . We define subsets T_{1}, T_{2} and T_{∗} of T as follows:

T_{1} = \{1\} ∪ \{ \binom{p}{c} | p, c ∈ ℤ, 1 ≤ p < c and gcd(p, c) = 1 \}
T_{2} = \{ ( \begin{matrix}p & 1 \\ c & a \end{matrix}) | a, p, c ∈ ℤ, a ≥ 1, 1 ≤ p < c and gcd(p, c) = 1 \}
T_{∗} = T_{1} ∪ T_{2}.

We will often identify the empty tableau with the column \binom{0}{1}. This allows us to write T_{1} = \{ \binom{p}{c} | p, c ∈ ℤ, 0 ≤ p < c and gcd(p, c) = 1 \}, which turns out to be very convenient. This practice is in agreement with convention 2.29 of [6]. Remark. Although the notations T_{1}, T_{2} and T_{∗} are not used in [5] and [6], the elements of those sets play an important role in the cited papers. In particular, T_{∗} is the set of tableaux satisfying one of the conditions (1–3) of [6]-2.16 (or equivalently one of the conditions (1–3) of [5]-5.6). 3.2. Definition. Given a tableau T = ( \begin{matrix}p_{1} & ··· & p_{k} \\ c_{1} & ··· & c_{k} \end{matrix}) ∈ T , define

μ(T) = ∏_{i=1}^{k} c_{i}

where in particular μ(1) = 1. The positive integer μ(T) is called the multiplicity of T. 3.3. Definition. Given T ∈ T_{∗}, define Ť ∈ T_{∗} by:

(1) If T = \binom{p}{c} ∈ T_{1}, then Ť = \binom{p′}{c} ∈ T_{1}, where p′ is defined by 0 ≤ p′ < c and pp′ ≡ 1 (mod c);

(2) If T = ( \begin{matrix}p & 1 \\ c & a \end{matrix}) ∈ T_{2}, then Ť = ( \begin{matrix}c−p & 1 \\ c & a \end{matrix}) ∈ T_{2}.

Note that ˇ1 = 1 and \binom{}{Ť}ˇ = T. 3.4. (See [6]-2.2 and [6]-2.3.) Let n be a positive integer. A weighted n-tuple is 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. If n = 1 then S is simply a weighted graph; if n = 2 then it is a weighted pair (2.9). Given x ∈ ℤ, let G_{(x)} be the weighted triple (G, v_{1}, v_{2}) where G is the weighted graph: [figure: 0; v_{1}; x; 0; v_{2}] .

3.5. Let S be a weighted n-tuple with n ≥ 2 and let T ∈ T be a tableau. By definitions 2.8–2.10 of [6], T may be regarded as a compact “recipe” for performing a finite sequence of blowings-up on S; the graph obtained at the end of this blowing-up process is denoted ST and is a weighted n-tuple (with the same n as S). If we delete from ST the vertex of weight −1 created by the last blowing-up of the sequence, we obtain a subgraph S ⊖ T of ST, and S ⊖ T is a weighted (n − 1)-tuple. One also defines subgraphs S [figure] T and S [figure] T of S ⊖ T, refer to [6]-2.10 for details. Note in particular:

• S ⊖ T is the disjoint union of S [figure] T and S [figure] T;

• S [figure] T is a weighted (n − 1)-tuple, with the same sequence of distinguished vertices as S ⊖ T.

• S [figure] T is a weighted graph; in fact, it is a (possibly empty) admissible chain. In particular, G_{(x)} ⊖ T and G_{(x)} [figure] T are weighted pairs with the same distinguished vertex, G_{(x)} [figure] T is a weighted graph (in fact an admissible chain) and G_{(x)} ⊖ T is the disjoint union of G_{(x)} [figure] T and G_{(x)} [figure] T. The following statement is an immediate consequence of [6]-2.12: 3.5.1. Lemma. If x ∈ ℤ and T ∈ T_{∗}, the weighted pair G_{(x)} ⊖ T is as follows.

If T = \binom{p}{c} ∈ T_{1}, then:

(23)  …  …^{0}_{∗}  ^{x}  _{p} _{c} \underbrace{G_{(x)} [figure] T}  \underbrace{G_{(x)} [figure] T} _{c} _{c−p}

If T = ( \begin{matrix}p & 1 \\ c & a \end{matrix}) ∈ T_{2}, then:

(24)  …  …^{0}_{∗}  ^{x}  _{acp+1} _{ac^{2}}  ^{−2}  ^{−2}  ^{−2}  ^{−2} \underbrace{G_{(x)} [figure] T}  \underbrace{G_{(x)} [figure] T} _{a}

In the above pictures, the distinguished vertex is marked with a “∗” and the integers under the braces are the determinants of the indicated linear chains (see [6]-1.10). Also, whenever the weight of a vertex is not indicated, that weight is strictly less than −1. 3.6. Various subsets of ℤ^{+} × T × T are considered in [5] and [6] (where ℤ^{+} denotes the set of positive integers). In particular (see [5]-5.24 or [6]-2.24):

𝕋_{0}(‡) = { (m, T_{1}, T_{2}) ∈ ℤ^{+} × T_{∗} × T_{∗} | each connected component of the

weighted graph (G_{(−m)} ⊖ T_{1}) ⊖ T_{2} is equivalent to an admissible chain}.

An explicit description of the elements of 𝕋_{0}(‡) is given in [5]-5.41. The set 𝕋_{0}(‡) is used for classifying affine rulings, as we will see in 3.8 and 3.9, below.

Affine rulings

We begin by defining the class of surfaces “satisfying the condition (‡).” By [6]-1.20 and 1.21, this class contains all weighted projective planes. 3.7. Consider the following condition on an algebraic surface X:

(†)

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. Also consider the stronger condition:

(‡)

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

If X is a complete normal rational surface, [6]-2.21 defines an affine ruling of X to be a linear system on X satisfying certain conditions. Suppose that X satisfies (‡).

• The notion of a basic affine ruling of X is defined in [6]-2.22.1.

• If Λ is an affine ruling of X then a nonempty subset Λ_{∗} of Λ is defined in

[6]-2.22.3. While writing the present paper we realised that statement 3.7.1 is true and can be

used as a definition of Λ_{∗}. 3.7.1. Let Λ be an affine ruling of a surface X satisfying (‡). Then, for any element F of Λ, the following conditions are equivalent:

(1) F ∈ Λ_{∗}

(2) Some element of the set G_{∞}[ X \ supp(F) ] is a chain of type (Z).

(3) Some element of the set G_{∞}[ X \ supp(F) ] is a linear chain.

In the same circle of ideas, we note: 3.7.2. Let Λ be an affine ruling of a surface X satisfying (‡), let F_{1}, …, F_{r} be distinct elements of Λ and let U = X \ supp(F_{1} + ⋯ + F_{r}). Then the following are equivalent:

(1) Some element of the set G_{∞}[U] is a chain of type (Z).

(2) r = 1 and F_{1} ∈ Λ_{∗}.

Facts 3.7.1 and 3.7.2 are implicitely contained in section 1 of [5], so we omit their proofs.

3.8. Consider triples (X, Λ, F) where X is a surface satisfying (‡), Λ is a basic affine ruling of X and F ∈ Λ_{∗}.

• Given a triple (X, Λ, F), [6]-2.23 defines an element disc(X, Λ, F) of 𝕋_{0}(‡), called the discrete part of (X, Λ, F). (See 3.6 for 𝕋_{0}(‡).)

• Two 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′. The equivalence class of (X, Λ, F) is denoted [X, Λ, F] and 𝕊_{0}(‡) is the set of equivalence classes.

• If (X, Λ, F) ∼ (X′, Λ′, F′), then (X, Λ, F) and (X′, Λ′, F′) have the same discrete part; so we may speak of the discrete part of the equivalence class [X, Λ, F] of (X, Λ, F), and we have a set map

disc : 𝕊_{0}(‡) → 𝕋_{0}(‡) [X, Λ, F] ↦ discrete part of [X, Λ, F].

The following may be regarded as a classification of triples (X, Λ, F): 3.9. Theorem. The map disc : 𝕊_{0}(‡) → 𝕋_{0}(‡) is bijective.

Proof. This is part of [5]-5.25. □

3.10. If X is a surface satisfying (‡) then [5]-5.2 defines a subset 𝕋_{0}(X) of 𝕋_{0}(‡) by

𝕋_{0}(X) = { disc(X, Λ, F) | Λ is a basic affine ruling of X and F ∈ Λ_{∗}}.

4. A result about tableaux

If a_{0}, a_{1}, a_{2} are pairwise relatively prime positive integers, [6]-2.26 defines four subsets

𝕋_{I}(a_{0}, a_{1}, a_{2}), 𝕋_{II.1}(a_{0}, a_{1}, a_{2}), 𝕋_{II.2}(a_{0}, a_{1}, a_{2}) and 𝕋_{III}(a_{0}, a_{1}, a_{2})

of 𝕋_{0}(‡). Results 4.2 and 4.3, below, are concerned with 𝕋_{I}(a_{0}, a_{1}, a_{2}), 𝕋_{II.1}(a_{0}, a_{1}, a_{2}) and 𝕋_{II.2}(a_{0}, a_{1}, a_{2}). 4.1. Lemma. Let a, b, c be positive integers such that c > 1 and gcd(b, c) = 1. Then there exists an integer y satisfying

by ≡ 1 (mod c), gcd(y, a) = 1 and 0 < y < ac.

Proof. Write a = αα′, where α and α′ are the positive integers satisfying

gcd(α, c) = 1

For every prime number p, p | α′ ⟹ p | c.

Since gcd(b, c) = 1, there exists an integer y_{0} satisfying by_{0} ≡ 1 (mod c). Since gcd(α, c) = 1, the Chinese Remainder Theorem implies that exactly one integer y satisfies 0 ≤ y < αc and

y ≡ y_{0} (mod c)
y ≡ 1 (mod α).

Then it is easy to see that y satisfies the desired conditions. □

4.2. Lemma. Given any T_{2} ∈ T_{2}, there exist relatively prime positive integers a_{0}, a_{1}, a_{2} and a tableau T_{1} ∈ T_{1} such that

(1, T_{1}, T_{2}) ∈ 𝕋_{II.1}(a_{0}, a_{1}, a_{2}).

Proof. We have T_{2} = ( \begin{matrix}p_{2} & 1 \\ c_{2} & a_{2} \end{matrix} ), where p_{2}, c_{2} and a_{2} are integers such that

c_{2} > p_{2} ≥ 1, gcd(p_{2}, c_{2}) = 1 and a_{2} ≥ 1.

LOG ℚ -HOMOLOGY PLANES AND WEIGHTED PROJECTIVE PLANES 19

By 4.1, we may choose an integer a_{1} satisfying (c_{2} − p_{2})a_{1} ≡ 1 (mod c_{2}) (25) gcd(a_{1}, a_{2}) = 1 (26) 0 < a_{1} < a_{2}c_{2}. (27) Then we may define an integer a_{0} by

(28)  a_{0} = a_{2}c_{2} − a_{1}.

Note that a_{0} > 0 by (27). We claim:

(29)  a_{0}, a_{1}, a_{2} are pairwise relatively prime positive integers.

In fact, we already know that gcd(a_{1}, a_{2}) = 1. From (25), we obtain gcd(a_{1}, c_{2}) = 1, so gcd(a_{1}, a_{2}c_{2}) = 1. Thus gcd(a_{0}, a_{1}) = gcd(a_{2}c_{2} − a_{1}, a_{1}) = gcd(a_{2}c_{2}, a_{1}) = 1, and gcd(a_{0}, a_{2}) = gcd(a_{2}c_{2} − a_{1}, a_{2}) = gcd(a_{1}, a_{2}) = 1. Hence, statement (29) is true.

Let k be the unique integer such that

(30)  \frac{c_{2}}{a_{1}} ≤ k < \frac{c_{2}}{a_{1}} + 1

and define integers x_{0}, x_{1} and x_{2} by

(31)  x_{1} = ka_{1} − c_{2}
(32)  x_{2} = \begin{cases}0, & if a_{2} = 1 \\ 1, & if a_{2} > 1. \end{cases}
(33)  x_{0} = k + x_{2} − 1.

Then, in the terminology of 7.2 of [6], we claim that

(34)  (x_{0}, x_{1}, x_{2}) is the unique solution to Eq(a_{0}, a_{1}, a_{2}).

Explicitely, (34) means that (x_{0}, x_{1}, x_{2}) satisfies the three conditions: a_{0} = a_{1}a_{2}x_{0} − a_{2}x_{1} − a_{1}x_{2} (35) 0 ≤ x_{1} < a_{1} (36) 0 ≤ x_{2} < a_{2}. (37) Indeed, (37) is obvious from (32), (36) follows from (31) and (30), and the calculation:

a_{1}a_{2}x_{0} − a_{2}x_{1} − a_{1}x_{2} = a_{1}a_{2}(k + x_{2} − 1) − a_{2}(ka_{1} − c_{2}) − a_{1}x_{2} by (33) and (31)
= a_{1}a_{2}(x_{2} − 1) + a_{2}c_{2} − a_{1}x_{2}
= a_{1}a_{2}(x_{2} − 1) + (a_{0} + a_{1}) − a_{1}x_{2}  by (28)
= a_{0} + a_{1}(a_{2} − 1)(x_{2} − 1) = a_{0}

shows that (35) holds. So (34) is true. We define two more integers by:

(38)  x′_{1} = \frac{a_{1}(c_{2} − p_{2}) − 1}{c_{2}}
(39)  x′′_{1} = kx′_{1} − c_{2} + p_{2}

Note that (25) implies that x′_{1} is an integer, so x′′_{1} is an integer as well. We claim:

(40) x′_{1} and x′′_{1} are the integers determined by the solution (x_{0}, x_{1}, x_{2}) to

Eq(a_{0}, a_{1}, a_{2}), as described in 7.2 of [6].

Explicitely, (40) means that the following conditions hold:

x_{1}x′_{1} ≡ 1 (mod a_{1}) and 0 ≤ x′_{1} < a_{1} (41)

|\begin{matrix}x_{1} & x′′_{1} \\ a_{1} & x′_{1} \end{matrix}| = 1. (42)

We check that these hold. Clearly, 0 ≤ x′_{1} < a_{1} follows immediately from (38). By (31) and (38),

x_{1}x′_{1} = (ka_{1} − c_{2})x′_{1} ≡ (−c_{2})x′_{1} = (−c_{2}) \frac{a_{1}(c_{2} − p_{2}) − 1}{c_{2}}

= 1 − a_{1}(c_{2} − p_{2}) ≡ 1 (mod a_{1}),

so (41) holds. By (31) and (39),

|\begin{matrix}x_{1} & x′′_{1} \\ a_{1} & x′_{1} \end{matrix}| = |\begin{matrix}ka_{1} − c_{2} & kx′_{1} − c_{2} + p_{2} \\ a_{1} & x′_{1} \end{matrix}| = |\begin{matrix}−c_{2} & −c_{2} + p_{2} \\ a_{1} & x′_{1} \end{matrix}| = |\begin{matrix}a_{1} & x′_{1} \\ c_{2} & c_{2} − p_{2} \end{matrix}| ,

and the last determinant is equal to 1 by (38), so (42) holds as well. So statement (40) is true. Next, we verify that

(43)  (\begin{matrix}p_{2} \\ c_{2} \end{matrix}) = (\begin{matrix}a_{1} − x_{1} − x′_{1} + x′′_{1} \\ a_{1} − x_{1} \end{matrix}) + (x_{0} − x_{2}) (\begin{matrix}a_{1} − x′_{1} \\ a_{1} \end{matrix}) .

To see this, note that the right hand side of (43) is equal to:

(\begin{matrix}(x_{0} − x_{2} + 1)(a_{1} − x′_{1}) − x_{1} + x′′_{1} \\ (x_{0} − x_{2} + 1)a_{1} − x_{1} \end{matrix}) = (\begin{matrix}k(a_{1} − x′_{1}) − x_{1} + x′′_{1} \\ ka_{1} − x_{1} \end{matrix})  by (33)

= (\begin{matrix}(ka_{1} − x_{1}) + (x′′_{1} − kx′_{1}) \\ ka_{1} − x_{1} \end{matrix})

= (\begin{matrix}c_{2} + (p_{2} − c_{2}) \\ c_{2} \end{matrix})  by (31) and (39)

so (43) is true.

Finally, we note that (28) gives \frac{a_{0}+a_{1}+a_{2}}{a_{2}} = \frac{a_{2}c_{2}+a_{2}}{a_{2}} = c_{2} + 1, hence:

(44)  \frac{a_{0} + a_{1} + a_{2}}{a_{2}} is a natural number strictly greater than 2.

We may now apply Proposition 7.4 of [6]. By (44) and the cited result, the set 𝕋_{II.1}(a_{0}, a_{1}, a_{2}) has exactly one element. The same result, together with (34), (40) and (43), implies that the unique element of 𝕋_{II.1}(a_{0}, a_{1}, a_{2}) is the triple:

(45)  (1, \binom{x′_{1}}{a_{1}}, (\begin{matrix}p_{2} & 1 \\ c_{2} & a_{2} \end{matrix})) .

This completes the proof of 4.2. □

Remark. In the above argument, a_{1} may be equal to 1 (this is the case iff c_{2} − p_{2} = 1). If a_{1} = 1, then x′_{1} = 0 and the column ( \begin{matrix}x′_{1} \\ a_{1} \end{matrix} ) = ( \begin{matrix}0 \\ 1 \end{matrix} ) which appears in (45) should be interpreted as the empty tableau. See 3.1. 4.3. Proposition. Given any T_{1} ∈ T_{∗}, there exist relatively prime positive integers a_{0}, a_{1}, a_{2} and a tableau T_{2} ∈ T_{1} such that

(1, T_{1}, T_{2}) ∈ 𝕋_{I}(a_{0}, a_{1}, a_{2}) ∪ 𝕋_{II.2}(a_{0}, a_{1}, a_{2}).

Proof. If T_{1} ∈ T_{1} then T_{1} = ( \begin{matrix}x_{1} \\ a_{1} \end{matrix} ) where x_{1}, a_{1} ∈ ℤ, 0 ≤ x_{1} < a_{1} and gcd(x_{1}, a_{1}) = 1. Define a_{0} = a_{1} − x_{1} and a_{2} = 1, then it is immediate that

a_{0}, a_{1}, a_{2} are pairwise relatively prime positive integers

and that

(1, x_{1}, 0) is the unique solution to Eq(a_{0}, a_{1}, a_{2})

(see (34–37) for meaning). Then Proposition 7.3 of [6] implies that

(1, T_{1}, 1) ∈ 𝕋_{I}(a_{0}, a_{1}, a_{2}),

so the assertion is true in this case.

If T_{1} ∈ T_{2} then, by 4.2, we have (1, T_{2}, T_{1}) ∈ 𝕋_{II.1}(a_{0}, a_{2}, a_{1}) for some tableau T_{2} ∈ T_{1} and some pairwise relatively prime positive integers a_{0}, a_{2}, a_{1}; then

(1, T_{1}, T_{2}) ∈ 𝕋_{II.2}(a_{0}, a_{1}, a_{2})

and we are done. □

We now give the geometric interpretation of 4.3. Recall from 3.10 that each surface X satisfying (‡) determines a subset 𝕋_{0}(X) of 𝕋_{0}(‡). In the case where X is a weighted projective plane, an explicit description of 𝕋_{0}(X) is given in [6]. In particular, [6]-7.1 implies that

𝕋_{I}(a_{0}, a_{1}, a_{2}) ∪ 𝕋_{II.2}(a_{0}, a_{1}, a_{2}) ⊆ 𝕋_{0}(ℙ(a_{0}, a_{1}, a_{2}))

for any pairwise relatively prime positive integers a_{0}, a_{1}, a_{2}. Thus result 4.3 has the following 4.4. Corollary. Given any T_{1} ∈ T_{∗}, there exist a weighted projective plane ℙ and a tableau T_{2} ∈ T_{1} such that

(1, T_{1}, T_{2}) ∈ 𝕋_{0}(ℙ).
5. The map 𝔉

5.1. Given a triple (X, Λ, F) as in 3.8, consider the open subset U = X \ supp(F) of X and use the affine ruling Λ to define a morphism ρ : U → 𝔸^{1} whose general fiber is an affine line (the fibers of ρ are the elements of the linear system Λ|_{U}). Although ρ is not unique, it is clear that the equivalence class [U, ρ] ∈ \overline{𝒩}^{+} (see 2.2) is uniquely determined by (X, Λ, F) and that, furthermore, [X, Λ, F] ↦ [U, ρ] is a well-defined map. We call this map the “restriction”, and write

\begin{matrix}𝕊_{0}(‡) & ⟶^{res} & \overline{𝒩}^{+} \\ [X, Λ, F] & ⟼ & [U, ρ]. \end{matrix}

We will see in 6.4 that the image of res is exactly \overline{ℳ}_{0}^{+}. 5.2. Recall from 3.9 that disc : 𝕊_{0}(‡) → 𝕋_{0}(‡) is bijective. We now consider the composition

𝕋_{0}(‡) ⟶^{disc^{−1}} 𝕊_{0}(‡) ⟶^{res} \overline{𝒩}^{+},

which we denote

𝔉 : 𝕋_{0}(‡) ⟶ \overline{𝒩}^{+}.

So the fact that 𝔉 is well-defined depends on 3.9, which is a nontrivial fact. 5.3. Proposition. Given two elements τ = (m, T_{1}, T_{2}) and τ′ = (m′, T′_{1}, T′_{2}) of 𝕋_{0}(‡),

𝔉(τ) = 𝔉(τ′) ⟺ T_{1} = T′_{1}.

Proof. Let τ = (m, T_{1}, T_{2}) ∈ 𝕋_{0}(‡) (we will not consider a τ′) and let [U, ρ] = 𝔉(τ). We will show that [U, ρ] is independent of T_{2} and m (see (49) and (50)), and this proves “⟸”. We will show that T_{1} can be recovered from [U, ρ] (see (57) and (58)), which proves “ ⟹ ”.

Let us explain how 𝔉 constructs [U, ρ] from τ. We consider each of the two steps:

τ ⟼^{disc^{−1}} [X, Λ, F] ⟼^{res} [U, ρ].

(i) The first step is to construct [X, Λ, F] from τ. This explanation is adapted from the more general [5]-5.29 (more general because 𝕋_{0}(‡) ⊂ 𝕋(†)). Consider the ruled surface 𝔽_{m}, its ruling Λ_{m} and directrix Σ_{m} ⊂ 𝔽_{m}. Choose a blowing-up

(46)  (X̃ →^{π} 𝔽_{m}, P_{1}, P_{2})

of 𝔽_{m} according to (T_{1}, T_{2}) (see [5]-5.26 for definition) and let Z_{1} and Z_{2} be the distinct elements of Λ_{m} satisfying P_{i} ∈ Z_{i}. In particular, recall that X̃ is a smooth projective surface, π is a birational morphism, cent(π) ⊆ {P_{1}, P_{2}} and for each i ∈ {1, 2} we have:

• π^{−1}(P_{i}) has at most one (−1)-component

• \overline{HN}(π, Z_{i}) = T_{i} (see [5]-4.1 for \overline{HN}).

For each i ∈ {1, 2}, define

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

Then E_{1}, E_{2} ⊂ X̃ are irreducible curves. Let Δ be the SNC-divisor of X̃ defined by

(47)  π^{−1}(Z_{1} ∪ Σ_{m} ∪ Z_{2}) = supp(E_{1} + Δ + E_{2}) and E_{1}, E_{2} ⊈ supp(Δ).

(Δ is called D in [5]-5.29, but D has a different meaning in the present argument). Then, as claimed in [5]-5.29, we have G(Δ, X̃) = (G_{(−m)} ⊖ T_{1}) ⊖ T_{2} and consequently Δ has a negative definite intersection matrix. So there exists a complete normal surface X and a birational morphism γ : X̃ → X with exc(γ) = supp(Δ). By [5]-5.29 we also obtain that X satisfies condition (†) of 3.7; that Λ_{m} determines an affine ruling Λ of X via the birational morphisms 𝔽_{m} ⟵^{π} X̃ ⟶^{γ} X; and that γ(E_{2}) is the support of an element F of Λ_{∗} (in [5]-5.29, F is called F_{2}).

In our special case (τ ∈ 𝕋_{0}(‡), so T_{2} ∈ T_{∗}), result [5]-5.11 implies that X satisfies (‡) and that Λ is a basic affine ruling of X; so we have constructed a triple (X, Λ, F) as in 3.8, i.e., [X, Λ, F] ∈ 𝕊_{0}(‡). Moreover, [5]-5.29 asserts that disc(X, Λ, F) = τ. (Remark: It is not trivial that [X, Λ, F] is independent of the choice of (46), but this is essentially the fact (3.9) that disc is injective.)

(ii) The second step is to define (U, ρ) from (X, Λ, F), by applying the restriction map 5.1. In particular, U is defined as

U = X \ C_{2},

where for each i ∈ {1, 2} we define the irreducible curve C_{i} = γ(E_{i}) ⊂ X.

We have explained how 𝔉 constructs [U, ρ] from τ. Now some remarks.

First, we describe the divisor Δ of X̃. As noted before, G(Δ, X̃) may be identified with (G_{(−m)} ⊖ T_{1}) ⊖ T_{2}. Now the weighted graph (G_{(−m)} ⊖ T_{1}) ⊖ T_{2} has connected components (G_{(−m)} [figure] T_{1}) [figure] T_{2}, G_{(−m)} [figure] T_{1} and (G_{(−m)} [figure] T_{1}) [figure] T_{2}, where the last two may be empty (see [6]-2.10 for ⊖, [figure] and [figure]). These correspond respectively to the three connected components 𝒞, 𝒜_{1} and 𝒜_{2} of supp(Δ), where 𝒞 is the one which contains the proper transform of Σ_{m} with respect to π, and where 𝒜_{1} and 𝒜_{2} are (possibly empty) admissible chains satisfying 𝒜_{i} ⊂ π^{−1}(P_{i}). Also, E_{i} meets 𝒞 and if 𝒜_{i} ≠ ∅ then E_{i} meets 𝒜_{i}.

In the above, we obtained U from X̃ by first shrinking 𝒜_{1}, 𝒞 and 𝒜_{2} and then removing C_{2} = γ(E_{2}) = γ(𝒞 ∪ E_{2} ∪ 𝒜_{2}). Alternatively, we obtain the same [U, ρ] if we shrink only 𝒜_{1} and then remove (the image of) 𝒞 ∪ E_{2} ∪ 𝒜_{2}. More precisely: Starting from X̃, let S be the normal surface obtained by shrinking 𝒜_{1} to a point and let σ : X̃ → S be the corresponding birational morphism. Then σ(𝒞 ∪ E_{2} ∪ 𝒜_{2}) is the support of an SNC-divisor D of S, and S \ supp(D) = U; let u : U ↪ S denote the inclusion map, then u is a smooth-normal compactification of U. Note that

(48)  π^{−1}(P_{2}) ⊂ π^{−1}(Σ_{m} ∪ Z_{2}) ⊂ 𝒞 ∪ E_{2} ∪ 𝒜_{2},

where the right-hand-side of (48) corresponds to the complement of U via σ. It follows that

(49)  [U, ρ] does not depend on T_{2}.

Indeed, by definition of a blowing-up of 𝔽_{m} according to (T_{1}, T_{2}), T_{2} only affects the part of π which could be described as the “blowing-up at points infinitely near P_{2}”. By (48), all that blowing-up occurs in the complement of U and so has no effect on [U, ρ]. So (49) is true. Similarly,

(50)  [U, ρ] does not depend on m.

This is because we can change the value of m by performing elementary operations: Blow-up 𝔽_{m} at a point of Z_{2} and then shrink the proper transform of Z_{2}. By (48), these operations affect only the complement of U, so (50) is true.

Note that (49) and (50) prove the implication “⟸” of 5.3. We now proceed to show that T_{1} is completely determined by [U, ρ]. By (49) and (50), [U, ρ] does not change if we replace (m, T_{1}, T_{2}) by (1, T_{1}, 1). So, from now-on, let us assume that

τ = (1, T_{1}, 1)

(where T_{1} is the same as before). First, we go over the definition of ρ, in a slightly different way. We begin the construction of the commutative diagram

(51)  [figure: 𝔸^{1}; ℙ^{1}; id; ℙ^{1}; 𝔽_{1}; X̃; U; S; σ; u; ρ; ρ̄; π; ρ_{1}]

by choosing a morphism ρ_{1} : 𝔽_{1} → ℙ^{1} compatible with the linear system Λ_{1} (i.e., the fibers of ρ_{1} are the elements of Λ_{1}). Since π shrinks exc(σ) = 𝒜_{1} to a point, there exists a morphism ρ̄: S → ℙ^{1} which makes the right hand side of (51) a commutative rectangle. Then simply define ρ to be the restriction of ρ̄. So we see that

(52)  \begin{matrix}U & ⟶^{u} & S \\ ^{ρ}↓ & & ↓ρ̄ \\ 𝔸^{1} & ⟶ & ℙ^{1} \end{matrix} is a completion of (U, ρ).

The boundary divisor of this completion is the D which we have already defined (see the paragraph before (48)). Moreover, if Σ̃_{1} ⊂ X̃ denotes the proper transform of Σ_{1} with respect to π, then σ(Σ̃_{1}) is the ρ-horizontal component of D. Let us now argue that we have the following isomorphisms of weighted graphs:

(53) G(D, S) ≅ G(𝒞 ∪ E_{2} ∪ 𝒜_{2}, X̃)

= G(𝒞 + Z̃_{2}, X̃) ≅ underlying weighted graph of G_{(−1)} [figure] T_{1},

and

(54)  G_{res}[U] ≅ G(𝒜_{1}, X̃) ≅ G_{(−1)} [figure] T_{1}.

First, G(D, S) is isomorphic to G(𝒞∪E_{2} ∪𝒜_{2}, X̃) because σ restricts to an isomorphism from an open neighborhood of 𝒞 ∪ E_{2} ∪ 𝒜_{2} to an open neighborhood of D. Then we have a genuine equality, G(𝒞 ∪ E_{2} ∪ 𝒜_{2}, X̃) = G(𝒞 + Z̃_{2}, X̃), because T_{2} = 1 implies that 𝒜_{2} = ∅ and that E_{2} = Z̃_{2}, where Z̃_{2} is the proper transform of Z_{2} with respect to π. We have already observed that G(Δ, X̃) may be identified with (G_{(−m)} ⊖ T_{1}) ⊖ T_{2} = (G_{(−m)} ⊖ T_{1}) ⊖ 1; since the effect of “⊖1” is to remove the distinguished vertex of G_{(−m)} ⊖ T_{1}, it follows that G(Δ + Z̃_{2}, X̃) may be identified with the underlying weighted graph of G_{(−1)} ⊖ T_{1}; or equivalently, that G(𝒞 + Z̃_{2}, X̃) and G(𝒜_{1}, X̃) may be respectively identified with the two graphs:

underlying weighted graph of G_{(−1)} [figure] T_{1}  and  G_{(−1)} [figure] T_{1}.

Finally, we may regard σ^{−1}(U) ⟶^{σ} U as the minimal resolution of singularities of U; since exc(σ) = 𝒜_{1}, G_{res}[U] may be identified with G(𝒜_{1}, X̃). This explains (53) and (54).

So the description of G_{(−1)} [figure] T_{1} and G_{(−1)} [figure] T_{1} given in 3.5.1 may be regarded as a description of G(D, S) and G_{res}[U]. This gives:

If T_{1} = \binom{p}{c} ∈ T_{1}, then:

(55)  [figure: …; …; σ(Z̃_{1}); 0; σ(Z̃_{2}); −1; σ(Σ̃_{1}); _{p}; _{c}; \underbrace{G(D, S)}; \underbrace{G_{res}[U]}; _{c}; _{c−p}]

If T_{1} = ( \begin{matrix}p & 1 \\ c & a \end{matrix}) ∈ T_{2}, then:

(56)  [figure: …; …; σ(Z̃_{1}); 0; σ(Z̃_{2}); −1; σ(Σ̃_{1}); _{acp+1}; _{ac^{2}}; −2; −2; −2; −2; \underbrace{G(D, S)}; \underbrace{G_{res}[U]}; _{a}]

In the above pictures, the integers under the braces are the determinants of the indicated linear chains and all weights which are not indicated are strictly less than −1. From (55) and (56), we see that σ(Σ̃_{1}) is the only (−1)-component of D; since this is the horizontal component, (52) is in fact a minimal completion of (U, ρ), so:

G_{∞}[U, ρ] = G(D, S)

(see 2.8 for G_{∞}[U, ρ]). From (55) and (56), it is obvious that G_{∞}[U, ρ] is a chain of type (Z) with bideterminant (see 2.16):

(57)  bidet G_{∞}[U, ρ] = \begin{cases}(c, p), & if T_{1} = \binom{p}{c} ∈ T_{1}; \\ (ac^{2}, acp + 1), & if T_{1} = ( \begin{matrix}p & 1 \\ c & a \end{matrix}) ∈ T_{2}. \end{cases}

It also follows that G_{res}[U] is an admissible chain with determinant:

(58)  det G_{res}[U] = \begin{cases}c, & if T_{1} = \binom{p}{c} ∈ T_{1}; \\ a, & if T_{1} = ( \begin{matrix}p & 1 \\ c & a \end{matrix}) ∈ T_{2}. \end{cases}

Observe that the first component of bidet G_{∞}[U, ρ] is equal to det G_{res}[U] if and only if T_{1} ∈ T_{1}. Thus, in view of (57) and (58), it is clear that T_{1} can be recovered from G_{∞}[U, ρ] and G_{res}[U]. So T_{1} is completely determined by [U, ρ], which proves the implication “ ⟹ ” of 5.3. □

The disjoint union of the weighted graphs G_{∞}[U, ρ] and G_{res}[U] occurred naturally in the above proof. 5.3.1. Definition. Given (U, ρ) ∈ 𝒩^{+}, define the weighted graph

G[U, ρ] = G_{∞}[U, ρ] ⊎ G_{res}[U]

where ⊎ means “disjoint union of weighted graphs”, and where the weighted graphs G_{res}[U] and G_{∞}[U, ρ] are defined in 2.1 and 2.8 respectively. Note that G[U, ρ] is uniquely determined by [U, ρ].

For future use, we record the geometric interpretation of the weighted pair G_{(−1)} ⊖T. 5.3.2. Lemma. Let T ∈ T_{∗} and let [U, ρ] = 𝔉(1, T, 1). Then G[U, ρ] has a unique vertex L of weight zero and

G_{(−1)} ⊖ T = (G[U, ρ], L).

More precisely, L belongs to the subgraph G_{∞}[U, ρ] of G[U, ρ] and we have:

G_{(−1)} [figure] T = (G_{∞}[U, ρ], L) and G_{(−1)} [figure] T = G_{res}[U].

Proof. Let T_{1} = T and note that τ = (1, T_{1}, 1) belongs to 𝕋_{0}(‡). Starting with this τ, go through the proof of 5.3. As noted in that proof, the underlying weighted graph of G_{(−1)} [figure] T_{1} is isomorphic to G(D, S) = G_{∞}[U, ρ]; comparing (23) with (55) and (24) with (56), we see that this isomorphism maps the distinguished vertex of G_{(−1)} [figure] T_{1} to σ(Z̃_{2}). So:

G_{(−1)} [figure] T_{1} = (G(D, S), σ(Z̃_{2})) = (G_{∞}[U, ρ], σ(Z̃_{2})).

It was also noted that G_{(−1)} [figure] T_{1} = G_{res}[U], so

G_{(−1)} ⊖ T_{1} = (G_{∞}[U, ρ] ⊎ G_{res}[U], σ(Z̃_{2})) = (G[U, ρ], σ(Z̃_{2})).

Looking at (55) and (56), we see that σ(Z̃_{2}) is the unique vertex of weight zero in G[U, ρ] = G(D, S) ⊎ G_{res}[U]; so σ(Z̃_{2}) is the L of the statement and we are done. □

5.4. Proposition. Let [U, ρ] be an element of the image of the map 𝔉. Then U ∈ ℳ_{0} and U is an open subset of a weighted projective plane.

Proof. We have [U, ρ] = 𝔉(τ′) for some τ′ = (m, T_{1}, T′_{2}) ∈ 𝕋_{0}(‡). Then T_{1} ∈ T_{∗} and, by 4.4, there exist a weighted projective plane ℙ and a tableau T_{2} ∈ T_{1} such that

(59)  (1, T_{1}, T_{2}) ∈ 𝕋_{0}(ℙ).

Write τ = (1, T_{1}, T_{2}), then 𝔉(τ) = [U, ρ] by 5.3. By (59) and the definition (3.10) of 𝕋_{0}(X), there exists a basic affine ruling Λ of ℙ and an element F of Λ_{∗} such that disc(ℙ, Λ, F) = τ. Recalling that 𝔉 is the composite

𝕋_{0}(‡) ⟶^{disc^{−1}} 𝕊_{0}(‡) ⟶^{res} \overline{𝒩}^{+},

we get [U, ρ] = 𝔉(τ) = res[ℙ, Λ, F] and in particular U ≅ ℙ \ supp(F). Then U is the complement of a curve in ℙ, so U is affine. More precisely, U = ℙ \ C_{2} (notation as in the proof of 5.3); since C_{2} is irreducible and, by [6]-1.7, Pic(ℙ_{s}) = ℤ (where ℙ_{s} is the smooth locus of ℙ), we have Pic(U_{s}) = ℤ/dℤ where d = deg(C_{2}), so Pic(U_{s}) is a finite group. (Remark: We have d = μ(T_{1}) by the remark following [5]-5.37. See 3.2 for the definition of μ.)

The fact that G_{∞}[U, ρ] is a chain of type (Z) was mentioned in the proof of 5.3, see in particular (57). Thus U satisfies conditions (a) and (b) of 2.20 and consequently ML(U) = k. Hence, U ∈ ℳ_{0}. □

6. The map 𝔣

6.1. Definition. If T ∈ T_{∗} then it is clear that (1, T, 1) ∈ 𝕋_{0}(‡) and 5.4 shows (in particular) that 𝔉(1, T, 1) ∈ \overline{ℳ}_{0}^{+}. So the following is a well-defined map:

\begin{matrix}𝔣 : T_{∗} & ⟶ & \overline{ℳ}_{0}^{+} \\ T & ⟼ & 𝔉(1, T, 1). \end{matrix}

By 5.3, 𝔣 is injective and has the same image as 𝔉. We will show that 𝔣 is bijective. Note that if U is any member of ℳ then there exists a morphism ρ : U → 𝔸^{1} whose

general fiber is an 𝔸^{1}, so the following paragraph applies to U. 6.2. Consider a pair (U, ρ) where U is a normal affine surface and ρ : U → 𝔸^{1} is a morphism whose general fiber is an 𝔸^{1}.

By 2.7, we may choose a minimal completion (3) of (U, ρ) whose horizontal component H satisfies H^{2} = −1; let D be the boundary divisor of that minimal completion. Let σ : X̃ → S be the minimal resolution of singularities of S and consider the commutative diagram (ignoring π for now):

(60)  \begin{matrix}U & ⟶^{u} & S & ⟵^{σ} & X̃ & ⟶^{π} & 𝔽_{1} \\ ^{ρ}↓ & & ↓ρ̄ \\ 𝔸^{1} & ⟶ & ℙ^{1} \end{matrix}

Let D̃ be the SNC-divisor of X̃ which satisfies supp (D̃) = σ^{−1}( supp(D)). Note that, since Sing(S) = Sing(U), σ restricts to an isomorphism going from an open neighborhood of D̃ to an open neighborhood of D. Let H̃ = σ^{−1}(H), then H̃^{2} = H^{2} = −1.

Let ρ̃ : X̃ → ℙ^{1} denote the composition X̃ ⟶^{σ} S ⟶^{ρ̄} ℙ^{1}. Since ρ̃ is a morphism whose general fiber is a ℙ^{1}, and since X̃ is smooth, rational and complete, we know that each reducible fiber of ρ̃ is a tree of projective lines which can be contracted to an irreducible curve, and that there exists such a contraction process which yields the surface 𝔽_{1}. More precisely, let us regard fibers of ρ̃ as divisors, i.e., we consider schemetheoretic fibers. Then each fiber F of ρ̃ satisfies F · H̃ = 1. Consequently, F has a unique component F^{∘} which meets H̃, and F^{∘} has multiplicity one in the divisor F. So we can write F = F^{∘} + F′, where the support of F′ is disjoint from H̃. Let F_{1}, …, F_{r} be the reducible fibers of ρ̃ and write F_{i} = F_{i}^{∘} + F′_{i} for every i. Then there exists a birational morphism π : X̃ → 𝔽_{1} whose exceptional locus is supp(F′_{1} + ⋯ + F′_{r}). This completes the definition of diagram (60).

For each i ∈ {1, …, r}, write F_{i} = 𝒟_{i} + 𝒜_{i} + Γ_{i}, where 𝒟_{i} (resp. 𝒜_{i}) contains all components of F_{i} which are part of D̃ (resp. which are part of exc(σ)), and where Γ_{i} contains all other components. Note that if C is a component of exc(σ) then ρ̃ shrinks C to a point; since C must be disjoint from D̃ (because Sing S = Sing U), we have C ∩ H̃ = ∅, so C ⊆ supp(F′_{i}) for some i and consequently:

exc(σ) = supp(𝒜_{1} + ⋯ + 𝒜_{r}) ⊆ exc(π).

From exc(σ) ⊆ exc(π), we infer that if some component C of exc(σ) satisfies C^{2} ≥ −1, then C^{2} = −1 and C is not a branching component of exc(σ); this is impossible, because we chose σ to be a minimal resolution of singularities. So:

(61)  Each component C of exc(σ) satisfies C^{2} < −1.

We claim:

(62) For each i ∈ {1, …, r}, supp(Γ_{i}) contains all (−1)-components of F_{i}.

Indeed, suppose that C is a component of F_{i} satisfying C^{2} = −1. Then C cannot be part of 𝒟_{i}, because the square in (60) is a minimal completion of (U, ρ); and it cannot be part of 𝒜_{i}, because of (61). So C must be in Γ_{i}, which proves (62). We claim:

(63)

For each i ∈ {1, …, r}, we have Γ_{i} ≠ 0 and 𝒟_{i} · H̃ = 1. Moreover, each

component C of Γ_{i} satisfies C ∩ H̃ = ∅ and C ∩ supp(𝒟_{i}) ≠ ∅.

Indeed, it is known that some component C_{∗} of F′_{i} satisfies C_{∗}^{2} = −1; by (62), C_{∗} is a component of Γ_{i}, so Γ_{i} ≠ 0. If C is any component of Γ_{i} then σ(C) is a complete curve, and hence is not contained in the affine surface U; thus C meets H̃ ∪ supp(𝒟_{i}). Applying this to C = C_{∗} gives C_{∗} ∩ supp(𝒟_{i}) ≠ ∅, because C_{∗} ∩ H̃ = ∅. In particular, 𝒟_{i} ≠ 0. Since U is affine, D̃ is connected, so H̃ ∪supp(𝒟_{i}) is connected and 𝒟_{i} ·H̃ > 0. Together with F_{i} · H̃ = 1, this implies that 𝒟_{i} · H̃ = 1 and that no component of Γ_{i} meets H̃, which proves (63).

Consider the fibers G_{1}, …, G_{s} of ρ̃ which are entirely contained in D̃. Note that each G_{i} is an irreducible curve such that G_{i}^{2} = 0 (otherwise G_{i} would have a (−1)- component, which is impossible because the square in (60) is a minimal completion of (U, ρ)). Also, we have s ≥ 1 because the composite U ⟶^{ρ} 𝔸^{1} ↪ ℙ^{1} is of course not surjective. Next we show:

(64)

Every branching component of supp (H̃ + ∑_{i=1}^{r} F_{i} + ∑_{i=1}^{s} G_{i}) is in fact a component of D̃.

To simplify language and notation, let us identify H̃ + ∑_{i=1}^{r} F_{i} + ∑_{i=1}^{s} G_{i} with the corresponding dual graph. By contradiction, suppose that C is a branch point of this graph and is not a vertex of D̃. For some i, C is a component of F_{i}; then C is a branch point of G_{1} + H̃ + F_{i}. Since G_{1} + H̃ + F_{i} shrinks to a linear chain, some branch ℬ of G_{1} + H̃ + F_{i} at C shrinks to the empty graph; since ℬ cannot contain the vertex G_{1} of weight zero, ℬ is included in F_{i}. We may consider a vertex C_{∗} of ℬ of weight −1; by (62) and (63), some vertex of 𝒟_{i} is adjacent to C_{∗}, so ℬ contains a vertex of 𝒟_{i}; since G_{1} + H̃ + 𝒟_{i} is a connected graph which does not contain C, it follows that ℬ contains the vertex G_{1}, a contradiction. This proves (64). Next, we have:

(65)

Let i ∈ {1, …, r} and let 𝒞 be a connected component of Γ_{i} + 𝒜_{i}. Then 𝒞 is a linear chain and exactly one component C of 𝒞 is a component of Γ_{i}. Moreover, C is a “terminal vertex” of the linear chain 𝒞.

The fact that 𝒞 is a linear chain follows immediately from (64). Since F_{i} is connected but 𝒜_{i} is disjoint from 𝒟_{i}, it follows that at least one vertex C of 𝒞 is a vertex of Γ_{i}. If C is not unique then, by (63), the connected set 𝒞 meets the connected set D̃ in at least two points, which is impossible because D̃ + 𝒞 is contained in a tree and hence has no loops. By (64), C is not a branch point of the graph 𝒟_{i} + C + 𝒜_{i}; however, some vertex of 𝒟_{i} is adjacent to C by (63); so at most one vertex of 𝒜_{i} is adjacent to C, which proves (65). From this and (61), it follows:

(66)

Each connected component of exc(σ) is an admissible chain. Thus every singular point of U is a cyclic quotient singularity.

Write Γ_{i} = ∑_{j=1}^{γ_{i}} c_{ij}C_{ij}, where C_{i1}, …, C_{i γ_{i}} are the distinct components of Γ_{i} and where the c_{ij} are positive integers. Let H_{i} denote the free abelian group generated by the symbols C_{i1}, …, C_{i γ_{i}} modulo the equation ∑_{j=1}^{γ_{i}} c_{ij}C_{ij} = 0. By a well-known argument,

(67)  Pic(U_{s}) = ⨁_{i=1}^{r} H_{i}.

In fact, let G denote a fiber of ρ̃. Since Pic(𝔽_{1}) is the free abelian group generated by a fiber and the directrix, Pic(X̃) is the free abelian group generated by G, H̃ and all components of exc(π). Now Pic(U_{s}) is Pic(X̃) modulo the subgroup generated by the components of D̃ and those of exc(σ) = supp(𝒜_{1} + ⋯ + 𝒜_{r}); this subgroup contains the linear equivalence class of G, because D̃ contains the fiber G_{1}. Also note that, by (63), we have supp(Γ_{i}) ⊆ exc(π) for every i ∈ {1, …, r}. These remarks and a straightforward calculation give (67). 6.2.1. If U ∈ ℳ then D̃ is a linear chain, s = 1 and r ≤ 1.

Proof. The fact that D̃ is a linear chain follows from 2.19, which also implies that s ≤ 1, hence s = 1. The number of vertices of (the dual graph of) D̃ which are adjacent to H̃ is at most two, and is precisely r + s (because 𝒟_{i} · H̃ = 1 for all i, by (63)), so r ≤ 1. □

6.2.2. If Pic(U_{s}) is a finite group then:

(1) For each i ∈ {1, …, r}, supp(Γ_{i}) is an irreducible curve (i.e., γ_{i} = 1).

(2) Each connected component of exc(π) has exactly one (−1)-component.

(3) exc(π) has r connected components, namely: F′_{1}, …, F′_{r}.

(4) For each i ∈ {1, …, r}, 𝒜_{i} is connected (and hence is an admissible chain).

(5) No two singular points of U belong to the same fiber of ρ.

Proof. Assertion (1) follows immediately from (67). Consider a connected component 𝒞 of exc(π). Since exc(π) = supp(F′_{1} + ⋯ + F′_{r}), and since distinct fibers of ρ̃ are disjoint, there exists i such that 𝒞 ⊆ F′_{i}. Now at least one component C of 𝒞 satisfies C^{2} = −1; by (62) we have C ⊆ supp(Γ_{i}), so in fact C = supp(Γ_{i}) by assertion (1). So 𝒞 has only one (−1)-component, which proves assertion (2); and if 𝒞 and 𝒞′ are connected components of exc(π) such that 𝒞 ∪ 𝒞′ ⊆ F′_{i}, then supp(Γ_{i}) ⊆ 𝒞 ∩ 𝒞′, so 𝒞 = 𝒞′ and assertion (3) is true. Assertion (4) follows from (65) and the fact that supp(Γ_{i}) is irreducible. Assertion (5) follows from (4). □

6.2.3. If U ∈ ℳ_{0} (and ρ is as before), there exists T ∈ T_{∗} such that 𝔣(T) = [U, ρ].

Proof. Consider the ruling Λ_{1} of 𝔽_{1} and its directrix Σ_{1} (and note that Σ_{1} = π(H̃)). Define distinct elements Z_{1}, Z_{2} of Λ_{1} and points P_{i} ∈ Z_{i} \ Σ_{1} (i = 1, 2) as follows:

• Set Z_{2} = π(G_{1}) and pick any point P_{2} of Z_{2} \ Σ_{1} (the curves G_{1}, …, G_{s} are defined before (64) and we have s = 1 by 6.2.1).

• If π is not an isomorphism then r = 1 by 6.2.1, so we may define Z_{1} and P_{1} by Z_{1} = π(F_{1}) and cent(π) = {P_{1}}; if π is an isomorphism, let Z_{1} be any element of Λ_{1} other than Z_{2} and let P_{1} be any point of Z_{1} \ Σ_{1}.

Then:

cent(π) ⊆ {P_{1}} and, for each i ∈ {1, 2}, π^{−1}(P_{i}) has at most one (−1)-component.

So, by [5]-4.1, the tableau T_{i} = \overline{HN}(π, Z_{i}) ∈ T is defined for each i ∈ \{1, 2\}. Note that T_{2} = 1 (the empty tableau), since cent(π) ∩ Z_{2} = ∅. By definition [5]-5.26, it is obvious that

(68)  (X̃ →^{π} 𝔽_{1}, P_{1}, P_{2}) is a blowing-up of 𝔽_{1} with respect to (T_{1}, T_{2}).

We now argue that T_{1} ∈ T_{∗}. It is a general fact that whenever (68) is true then so is

(69)  G(Δ, X̃) = (G_{(−1)} ⊖ T_{1}) ⊖ T_{2},

where Δ is the SNC-divisor of X̃ defined exactly as in (47), in the proof of 5.3. In fact the definition of Δ implies:

supp(Δ) = \begin{cases}supp(H̃ + 𝒟_{1}) ∪ supp(𝒜_{1}), & if π is not an isomorphism, \\ H̃, & if π is an isomorphism \end{cases}

and recall that 𝒜_{1} is an admissible chain by 6.2.2. So in all cases we have supp(Δ) = 𝒞∪𝒜, where 𝒞 and 𝒜 are disjoint, 𝒞 ⊆ supp(D̃) and 𝒜 is a (possibly empty) admissible chain. Since D̃ is a linear chain by 6.2.1, 𝒞 is a linear chain and Δ satisfies: Every connected component of Δ is a linear chain and every (irreducible) component C of Δ other than H̃ satisfies C^{2} < −1. By (69), this translates into

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

Then part (2) of result [5]-5.7 implies that

T_{1} ∈ T_{∗}.

Let τ = (1, T_{1}, T_{2}) = (1, T_{1}, 1) and note that τ ∈ 𝕋_{0}(‡). Then we claim that 𝔉(τ) = [U, ρ]. Indeed, this is clear from consideration of the beginning of the proof of 5.3, up to (48) (the notation is compatible). Thus 𝔣(T_{1}) = [U, ρ]. □

6.3. Theorem. The map 𝔣 : T_{∗} → \overline{ℳ}_{0}^{+} is bijective.

Proof. 𝔣 is injective by 5.3 and surjective by 6.2.3. □

6.4. Corollary. The three maps

res : 𝕊_{0}(‡) ⟶ \overline{𝒩}^{+}, 𝔉 : 𝕋_{0}(‡) ⟶ \overline{𝒩}^{+} and 𝔣 : T_{∗} ⟶ \overline{ℳ}_{0}^{+}

have the same image, namely, \overline{ℳ}_{0}^{+}.

Proof. Immediate. □

6.5. Theorem. Let U and U′ be surfaces belonging to the class ℳ_{0}. If G_{∞}[U] = G_{∞}[U′] and G_{res}[U] = G_{res}[U′] then U ≅ U′.

Proof. By 2.20 and 2.17, we may pick an element of G_{∞}[U] of the form

(70)  [figure: …; 0; −1; ω_{1}; ω_{q}]

where q ≥ 0 and ∀_{i} ω_{i} ≤ −2. Consequently, we can find a smooth-normal compactification u : U ↪ S of U such that the weighted graph G(u) is identical to (70). Let L be the unique 0-component of S \ U. Then the complete linear system |L| on S is one-dimensional, free of base points and has the following property:

The general member G ∈ |L| satisfies G ≅ ℙ^{1} and G ∩ U = 𝔸^{1}.

We may consider a morphism ρ̄: S → ℙ^{1} whose fibers are the elements of |L|. Restricting ρ̄ to U gives a morphism ρ : U → 𝔸^{1} whose general fiber is an 𝔸^{1}, i.e., (U, ρ) ∈ ℳ_{0}^{+}. Then the commutative diagram

\begin{matrix}U & ⟶^{u} & S \\ ^{ρ}↓ & & ↓ρ̄ \\ 𝔸^{1} & ⟶ & ℙ^{1} \end{matrix}

is a minimal completion of (U, ρ) and consequently the weighted graph G_{∞}[U, ρ] (defined in 2.8) is (70).

Since G_{∞}[U] = G_{∞}[U′], the weighted graph (70) is also an element of G_{∞}[U′]; so the above argument shows that there exists a morphism ρ′ : U′ → 𝔸^{1} such that (U′, ρ′) ∈ ℳ_{0}^{+} and such that G_{∞}[U′, ρ′] is (70).

Since 𝔣 is bijective, we may consider T, T′ ∈ T_{∗} such that f(T) = [U, ρ] and f(T′) = [U′, ρ′]; then by 5.3.2

G_{(−1)} ⊖ T = (G[U, ρ], L) and G_{(−1)} ⊖ T′ = (G[U′, ρ′], L′)

where L (resp. L′) is the unique vertex of weight zero in G[U, ρ] (resp. in G[U′, ρ′]). Since both G_{∞}[U, ρ] and G_{∞}[U′, ρ′] are identical to (70), and since G_{res}[U] = G_{res}[U′] by assumption, it follows that G[U, ρ] is identical to G[U′, ρ′]. So

G_{(−1)} ⊖ T = G_{(−1)} ⊖ T′.

By [5]-5.14, it follows that T = T′. Thus [U, ρ] = 𝔣(T) = 𝔣(T′) = [U′, ρ′] and in particular U ≅ U′. □

The map 𝔣̄

6.6. Definition. Let 𝔣̄: T_{∗} → \overline{ℳ}_{0} be the composition

T_{∗} ⟶^{𝔣} \overline{ℳ}_{0}^{+} ⟶^{proj} \overline{ℳ}_{0}

where the projection map \overline{ℳ}_{0}^{+} ⟶^{proj} \overline{ℳ}_{0} was defined in 2.2. Note that 𝔣̄ is surjective. 6.7. Theorem. The inverse image by 𝔣̄ of any element of \overline{ℳ}_{0} is \{T, Ť\} for some T ∈ T_{∗}.

Proof. We first show that each T ∈ T_{∗} satisfies 𝔣̄(T) = 𝔣̄(Ť). Let [U, ρ] = 𝔣(T) and [U′, ρ′] = 𝔣(Ť). By 5.3.2, G_{(−1)} ⊖ T = (G[U, ρ], L) and G_{(−1)} ⊖ Ť = (G[U′, ρ′], L′), where L (resp. L′) is the unique vertex of weight zero in G[U, ρ] (resp. in G[U′, ρ′]). Using the notation of [5]-3.36 and result [5]-5.14,

(G[U′, ρ′], L′) = G_{(−1)} ⊖ Ť = (G_{(−1)} ⊖ T )^{t} = (G[U, ρ], L)^{t}.

Whenever 𝒫 and 𝒫′ are weighted pairs satisfying 𝒫′ = 𝒫^{t}, the underlying weighted graphs of 𝒫 and 𝒫′ are equivalent. Thus G[U, ρ] and G[U′, ρ′] are equivalent weighted graphs; it easily follows that G_{∞}[U, ρ] ∼ G_{∞}[U′, ρ′] and G_{res}[U] ∼ G_{res}[U′]. Since G_{∞}[U, ρ] ∈ G_{∞}[U] and G_{∞}[U′, ρ′] ∈ G_{∞}[U′], we get G_{∞}[U] = G_{∞}[U′]; since G_{res}[U] and G_{res}[U′] are equivalent admissible chains, they are identical. Then 6.5 gives U ≅ U′, hence 𝔣̄(T) = [U] = [U′] = 𝔣̄(Ť).

Conversely, consider T, T′ ∈ T_{∗} such that 𝔣̄(T) = 𝔣̄(T′); we show that T′ ∈ {T, Ť}. Let [U, ρ] = 𝔣(T) and [U, ρ′] = 𝔣(T′) (with the same U). By 5.3.2, G_{(−1)} ⊖ T = (G[U, ρ], L) and G_{(−1)} ⊖ T′ = (G[U, ρ′], L′), where L (resp. L′) is the unique vertex of weight zero in G[U, ρ] (resp. in G[U, ρ′]). By 2.19, the connected component G_{∞}[U, ρ] of G[U, ρ] has the form

(71)  [figure: …; 0; −1; ω_{1}; ω_{q}]

where q ≥ 0 and (for all i) ω_{i} ≤ −2. Since G_{∞}[U, ρ] and G_{∞}[U, ρ′] belong to G_{∞}[U], they are equivalent and 2.17 implies that G_{∞}[U, ρ′] is either (71) or

(72)  [figure: …; 0; −1; ω_{q}; ω_{1}]

Consequently, (G[U, ρ′], L′) is either (G[U, ρ], L) or (G[U, ρ], L)^{t}. In the first case, we have

G_{(−1)} ⊖ T′ = (G[U, ρ′], L′) = (G[U, ρ], L) = G_{(−1)} ⊖ T,

and [5]-5.14 gives T′ = T; in the second case,

G_{(−1)} ⊖ T′ = (G[U, ρ′], L′) = (G[U, ρ], L)^{t} = (G_{(−1)} ⊖ T )^{t} = G_{(−1)} ⊖ Ť,

and [5]-5.14 gives T′ = Ť. So T′ ∈ {T, Ť}. □

6.8. Corollary. Let T ∈ T_{∗}, let [U] = 𝔣̄(T) and write U = Spec R. Consider the action of the group Aut_{k}(R) on the set klnd(R) (see the introduction). Then:

(1) The number of orbits is either one or two.

(2) The action is transitive if and only if T = Ť.

Proof. Consider the inverse image E ⊂ \overline{ℳ}_{0}^{+} of [U] ∈ \overline{ℳ}_{0} by the projection \overline{ℳ}_{0}^{+} → \overline{ℳ}_{0}. Then by definition 6.6 of 𝔣̄ we have 𝔣^{−1}(E) = 𝔣̄^{−1}([U]), which is equal to {T, Ť} by 6.7. So, by 6.3, 𝔣 restricts to a bijection {T, Ť} → E. The result follows from 2.4, which gives a bijection between E and the set of orbits. □

The following gathers some useful facts concerning the surjection 𝔣̄: T_{∗} → \overline{ℳ}_{0}. 6.9. Corollary. Let T ∈ T_{∗} and let [U] = 𝔣̄(T).

(1) The Picard group of U_{s} = U \ Sing(U) is ℤ/μ(T)ℤ.

(2) G_{res}[U] = G_{(−1)} [figure] T

(3) The underlying weighted graph of the weighted pair G_{(−1)} [figure] T is an element of G_{∞}[U].

(4) The surface U is smooth if and only if T = 1 or T = ( \begin{matrix}p & 1 \\ c & 1 \end{matrix} ) where c > p ≥ 1 and gcd(p, c) = 1.

(5) U is symmetric at infinity if and only if Ť = T (symmetry at infinity is defined in the introduction).

(6) If T = 1, then U = 𝔸^{2}; if T = ( \begin{matrix}1 & 1 \\ 2 & 1 \end{matrix} ), then U is ℙ^{2} minus a conic. These are the only members of ℳ_{0} which are smooth and symmetric at infinity.

Proof. The point of this result is to have a statement which does not involve a morphism ρ : U → 𝔸^{1}. However, the proof consists in applying earlier results to [U, ρ] = 𝔣(T).

Assertion (1) was noted in the proof of 5.4.

Assertions (2) and (3) reiterate 5.3.2.

By assertion (2), U is smooth iff G_{(−1)} [figure] T is the empty graph, which is the case iff the entry of T in the lower right position is a 1. This proves assertion (4).

To prove assertion (5), note that U is symmetric at infinity iff (G[U, ρ], L)^{t} = (G[U, ρ], L). By 5.3.2, this is equivalent to (G_{(−1)} ⊖ T )^{t} = G_{(−1)} ⊖ T; and by [5]-5.14, this is equivalent to Ť = T.

We leave the verification of (6) to the reader. □

To complement assertions (2) and (3) of 6.9, note that the graphs G_{(−1)} [figure] T and G_{(−1)} [figure] T are explicitely described in 3.5.1. 6.10. Corollary. If U ∈ ℳ_{0} is singular, then U has only one singular point and it is a cyclic quotient singularity.

Proof. This follows from part (2) of 6.9, because G_{(−1)} [figure] T is an admissible chain. □

If c ≥ 2 is an integer, let φ(c) denote the number of integers x satisfying gcd(x, c) = 1 and 0 < x < c. Then: 6.11. Corollary. Let c > 2 be an integer. Then, up to isomorphism, there exist exactly φ(c)/2 smooth surfaces in ℳ_{0} having a Picard group of order c.

Proof. Follows immediately from 6.9 and 6.7. □

7. Embeddings of surfaces

7.1. Theorem. Given T ∈ T_{∗}, define (a_{0}, a_{1}, a_{2}) ∈ ℤ^{3} and f ∈ k[X_{0}, X_{1}, X_{2}] as follows. (1) If T = \binom{p}{c} ∈ T_{1}, let (a_{0}, a_{1}, a_{2}) = (c − p, c, 1) and f = X_{1}.

(2) If T = ( \begin{matrix}p & 1 \\ c & a \end{matrix}) ∈ T_{2}, there exists a_{2} ∈ ℤ satisfying:

(c − p)a_{2} ≡ 1 (mod c), gcd(a_{2}, a) = 1 and 0 < a_{2} < ac.

Choose any such a_{2} and define a_{0} = ac − a_{2}, a_{1} = a and f = X_{0}X_{2} + X_{1}^{c}. Then a_{0}, a_{1}, a_{2} are pairwise relatively prime positive integers and f is homogeneous with respect to the grading of k[X_{0}, X_{1}, X_{2}] given by deg X_{i} = a_{i} (i = 0, 1, 2). Moreover, the surface 𝔣̄(T) is the complement of the curve f = 0 in ℙ(a_{0}, a_{1}, a_{2}).

Proof. It is immediate that a_{0}, a_{1}, a_{2} are pairwise relatively prime positive integers and that f is homogeneous with respect to the given grading. Write R = k[X_{0}, X_{1}, X_{2}] and always regard R as a graded algebra (so an “automorphism of R” is an automorphism of graded algebra). Let ℙ = ℙ(a_{0}, a_{1}, a_{2}) and, given a homogeneous element g ∈ R, let V (g) ⊂ ℙ denote the zero-set of g.

The proofs of 4.3 and 4.2 tell us that, if we define (a_{0}, a_{1}, a_{2}) as in the above statement, then there exists a tableau T_{2} ∈ T_{1} such that

(1, T, T_{2}) ∈ \begin{cases}𝕋_{I}(a_{0}, a_{1}, a_{2}) & if T ∈ T_{1} \\ 𝕋_{II.2}(a_{0}, a_{1}, a_{2}) & if T ∈ T_{2}. \end{cases}

Write τ = (1, T, T_{2}).

Consider the case where T = \binom{p}{c} ∈ T_{1}; by [6]-7.3 and [6]-3.1, the set 𝕋_{I}(a_{0}, a_{1}, a_{2}) is a singleton and its unique element is disc[ℙ, Λ_{0}, F], where Λ_{0} is a basic affine ruling of ℙ defined in [6]-3.1 and F is an element of (Λ_{0})_{∗} satisfying supp(F) = V (X_{1}). So, by definition of 𝔉, we have 𝔉(τ) = res[ℙ, Λ_{0}, F] = [ℙ \ V (X_{1}), ρ] for some ρ : ℙ \ V (X_{1}) → 𝔸^{1}. Since 𝔣(T) = 𝔉(τ), we obtain 𝔣̄(T) = [ℙ \ V (X_{1})].

Consider the other case: T = ( \begin{matrix}p & 1 \\ c & a \end{matrix}) ∈ T_{2}. By [6]-7.1 we have 𝕋_{II.2}(a_{0}, a_{1}, a_{2}) ⊂ 𝕋_{0}(ℙ) so τ = disc(ℙ, Λ, F) for some basic affine ruling Λ of ℙ and some element F of Λ_{∗}. As in the first case, it follows that 𝔣̄(T) = [ℙ \ supp F]. Since every element of Λ has irreducible support by [5]-1.8, supp(F) is an irreducible curve C ⊂ ℙ; then deg C = μ(T) = ac = a_{1}c by 6.9. Let f be a homogeneous prime element of R such that V (f) = C. To finish the proof it suffices to show that

(73)  θ(f) = X_{0}X_{2} + X_{1}^{c}, for some automorphism θ of the graded algebra R.

By [5]-1.15, Λ has at most two multiple members; so we may pick F′ ∈ Λ such that F′ ≠ F and {F, F′} contains all multiple members of Λ. Let f′ be a homogeneous prime element of R such that supp(F′) = V (f′). We claim:

(74)  θ′(f′) = X_{2}, for some automorphism θ′ of the graded algebra R.

To see this, note that the fact that Λ is a basic affine ruling of ℙ immediately implies that F′ ∈ Λ_{∗} and disc(ℙ, Λ, F′) = (1, T_{2}, T) (see part (2) of [6]-2.26, for instance). Since (1, T, T_{2}) ∈ 𝕋_{II.2}(a_{0}, a_{1}, a_{2}), it is obvious that (1, T_{2}, T) ∈ 𝕋_{II.1}(a_{0}, a_{2}, a_{1}); then [6]-5.1 asserts that (1, T_{2}, T) is not minimal in 𝕋(‡) and that its immediate predecessor (call it τ^{−}) is some element of 𝕋_{I}(a_{0}, a_{2}, a_{1}); thus

disc(ℙ, Λ, F′) > τ^{−} ∈ 𝕋_{I}(a_{0}, a_{2}, a_{1}).

By [5]-5.13, there exists an affine ruling Λ^{−} of ℙ and an element F^{−} of Λ_{∗}^{−} such that disc(ℙ, Λ^{−}, F^{−}) = τ^{−} and supp(F^{−}) = supp(F′). On the other hand, we obtain (by [6]- 7.3 and [6]-3.1) that the set 𝕋_{I}(a_{0}, a_{2}, a_{1}) is a singleton and that its unique element τ^{−} is disc(ℙ, Λ_{0}, F′′), where supp(F′′) = V (X_{2}). Thus disc(ℙ, Λ_{0}, F′′) = disc(ℙ, Λ^{−}, F^{−}) and by 3.9 we obtain [ℙ, Λ_{0}, F′′] = [ℙ, Λ^{−}, F^{−}]. In particular, there exists an automorphism of ℙ which carries supp(F′′) = V (X_{2}) onto supp(F^{−}) = supp(F′). This proves (74).

So we may assume that f′ = X_{2}. Using [5]-1.15 again, we obtain that ℙ \ (supp(F) ∪ supp(F′)) is isomorphic to ℙ^{2} minus two lines. Since gcd(deg(f), deg(f′)) = gcd(a_{1}c, a_{2}) = 1, Theorem 3.5 of [4] implies that

k[X_{2}, f] = ker D,

for some homogeneous locally nilpotent derivation D : R → R. Here, we may choose D in such a way that D(R) ⊈ X_{2}R; then D̄ ≠ 0, where D̄ : R̄ → R̄ is the locally nilpotent derivation “D mod X_{2}”, R̄ = R/X_{2}R ≅ k[X_{0}, X_{1}]. By Rentschler’s Theorem [12], there exist u, v such that ker(D̄) = k[v] and k[u, v] = k[X_{0}, X_{1}]; moreover, v is homogeneous.

Note that deg(f) = a_{1}c = a_{0}+a_{2} > a_{2}, so deg(f) > deg(X_{2}). Since f is irreducible, it follows that X_{2} |̸ f. So f(X_{0}, X_{1}, 0) is a nonconstant element of ker D̄ and consequently

(75)

There exists a homogeneous variable v of k[X_{0}, X_{1}] such that f(X_{0}, X_{1}, 0) is a power of v.

We may write f = g(X_{0}, X_{1}) + X_{2}H(X_{0}, X_{1}, X_{2}) where g ∈ k[X_{0}, X_{1}] and H ∈ k[X_{0}, X_{1}, X_{2}] are homogeneous. We have g = v^{n} by (75), so f = v^{n}+X_{2}H(X_{0}, X_{1}, X_{2}).

Given p, q ∈ R, let p ∼ q mean that p, q are associates. Note that v ∼ X_{0} + λX_{1}^{m} or v ∼ X_{1} + λX_{0}^{m} (where λ ∈ k and m > 0). We have k[v, X_{1}, X_{2}] = R in the first case and k[X_{0}, v, X_{2}] = R in the second case. So we may assume that one of the following holds:

(i) f = X_{0}^{n} + X_{2}H(X_{0}, X_{1}, X_{2}), or (ii) f = X_{1}^{c} + X_{2}H(X_{0}, X_{1}, X_{2}).

Note that H ≠ 0 by irreducibility of f, so deg(H) = a_{0}.

In case (i), we have a_{0} | deg(f) = a_{1}c = a_{0} + a_{2}, so a_{0} | a_{2} and a_{0} = 1. So deg(H) = 1 and in particular H is a linear form in X_{0}, X_{1}, X_{2}. If H ∈ k[X_{0}, X_{2}], then f is a homogeneous prime element of k[X_{0}, X_{2}] and consequently f ∼ X_{0} or f ∼ X_{2} or f ∼ X_{0}^{a_{2}} + λX_{2}^{a_{0}} = X_{0}^{a_{2}} + λX_{2} (some λ ∈ k^{∗}); this is absurd because deg(f) = a_{0} + a_{2} > max(a_{0}, a_{2}), so we conclude that H ∉ k[X_{0}, X_{2}]. Thus k[X_{0}, H, X_{2}] = R, so f = X_{0}^{n} + X_{2}X_{1} up to an automorphism. It follows that a_{1} = a_{0} = 1 and that n = c, so the interchange of X_{0} and X_{1} is an automorphism and gives (73).

Consider (ii). If H ∈ k[X_{1}, X_{2}], then f is a homogeneous prime element of k[X_{1}, X_{2}], so f ∼ X_{1} or f ∼ X_{2} or f ∼ X_{1}^{a_{2}} + λX_{2}^{a_{1}} (some λ ∈ k^{∗}); since deg(f) = a_{1}c = a_{0} + a_{2} and c > 1, we have deg(f) > max(a_{1}, a_{2}), so we may rule out the first two cases and conclude that f ∼ X_{1}^{a_{2}} +λX_{2}^{a_{1}} (some λ ∈ k^{∗}); then a_{2} = c, which is impossible because c > 1 and gcd(c, a_{2}) = 1. So H ∉ k[X_{1}, X_{2}]; since deg(H) = a_{0}, we deduce that H ∼ X_{0} + g(X_{1}, X_{2}) for some g ∈ k[X_{1}, X_{2}]; then k[H, X_{1}, X_{2}] = R and, up to an automorphism, f = X_{1}^{c} + X_{2}X_{0}, i.e., (73) holds and 7.1 is proved. □

We now address the questions:

• Which open subsets of ℙ(a_{0}, a_{1}, a_{2}) are members of ℳ_{0}?

• Given U ∈ ℳ_{0} and (a_{0}, a_{1}, a_{2}), decide whether U can be embedded in ℙ(a_{0}, a_{1}, a_{2}). Recall that all weighted projective planes belong to the class of surfaces satisfying the condition (‡) (see [6]-1.20 and 1.21). 7.2. Definition. Let X be a surface satisfying (‡). A special curve of X is an irreducible curve C ⊂ X satisfying:

There exists an affine ruling Λ of X and a positive integer n such that nC ∈ Λ_{∗}. Remark. Let Λ be an affine ruling of a surface X satisfying (‡). Then, by [5]-1.8, every element of Λ has irreducible support. So, given any F ∈ Λ_{∗}, supp(F) is a special curve of X. 7.2.1. Lemma. Let C be a special curve of a surface X satisfying (‡). Then there exists a basic affine ruling Λ of X and a positive integer n such that nC ∈ Λ_{∗}.

Proof. Let Λ be an affine ruling of X such that nC ∈ Λ_{∗} for some n > 0. Set (X_{1}, Λ_{1}, F_{1}) = (X_{2}, Λ_{2}, F_{2}) = (X, Λ, nC) and apply [5]-5.34. Part (1) of that result gives a basic affine ruling Λ′_{1} of X and an element F′_{1} of (Λ′_{1})_{∗} satisfying supp(F′_{1}) = supp(nC). □

7.3. Proposition. For an open subset U of a surface X satisfying (‡), the following conditions are equivalent:

(1) U ∈ ℳ

(2) U ∈ ℳ_{0}

(3) X \ U is a special curve of X.

Proof. We first show that (3) implies (2). Suppose that C = X \ U is a special curve of X. By 7.2.1, there exists a basic affine ruling Λ of X and an element F of Λ_{∗} such that supp(F) = C. Then [X, Λ, F] ∈ 𝕊_{0}(‡) and consequently we may evaluate the map res : 𝕊_{0}(‡) → \overline{𝒩}^{+} at [X, Λ, F] (see 5.1). It is immediate that

res[X, Λ, F] = [U, ρ]

for some ρ : U → 𝔸^{1}. By 6.4 we conclude that U ∈ ℳ_{0}, so (2) holds.

It is obvious that (2) implies (1). We prove that (1) implies (3). Assume that U ∈ ℳ. Then there exists a morphism ρ : U → 𝔸^{1} whose general fiber is an affine line. By known properties of such morphisms, there holds:

(76)

There exists a nonempty open subset V of 𝔸^{1} such that ρ^{−1}(V) ≅ V × 𝔸^{1} and such that the composition V × 𝔸^{1} → ρ^{−1}(V) ⟶^{ρ} V is the projection.

Now ρ extends to a rational map X −−→ ℙ^{1} which, in turn, determines a linear system Λ on X whose base locus is a finite subset of X \ U. By (76), Λ is an affine ruling of X and there exists a finite subset {M_{1}, … M_{r}} of Λ satisfying

(77)  X \ supp(M_{1} + ⋯ + M_{r}) ⊆ U

(the existence of the set {M_{1}, … M_{r}} is asserted, e.g., by [5]-1.11). Note that, by [5]- 1.8, every element of Λ has irreducible support; also, X \ U is a nonempty union of curves, because U is affine and X is normal and complete. Consequently, if we choose the set {M_{1}, … M_{r}} to be minimal with respect to property (77), then:

(78) Equality holds in (77).

By 2.20, the assumption that U ∈ ℳ implies that some element of the set G_{∞}[U] is a chain of type (Z). Then (78) and 3.7.2 give r = 1 and M_{1} ∈ Λ_{∗}, so (3) holds. □

7.4. Definition. Given a surface X satisfying (‡), we define a subset 𝕋_{□}(X) of T_{∗} by:

𝕋_{□}(X) = { T ∈ T_{∗} | ∃_{(m,T′)∈ℤ^{+}×T_{∗}} such that (m, T, T′) ∈ 𝕋_{0}(X)}.

See 3.10 for the definition of 𝕋_{0}(X). Remark: The following is easy to see: (m, T, T′) ∈ 𝕋_{0}(X) ⟺ (m, T′, T) ∈ 𝕋_{0}(X).

7.5. Theorem. Let U ∈ ℳ_{0} and let T ∈ T_{∗} be such that 𝔣̄(T) = [U]. Given a surface X satisfying (‡), the following are equivalent:

(1) U is isomorphic to some open subset of X

(2) {T, Ť} ∩ 𝕋_{□}(X) ≠ ∅.

Proof. Suppose that (1) holds. Then, by 7.3, U = X \ C where C is a special curve of X. By 7.2.1, there exists a basic affine ruling Λ of X and an element F of Λ_{∗} such that supp(F) = C. It is clear that [X, Λ, F] ∈ 𝕊_{0}(‡) and that res[X, Λ, F] = [U, ρ] for some ρ : U → 𝔸^{1}. So, if we define (m, T_{1}, T_{2}) = disc[X, Λ, F], we have T_{1} ∈ 𝕋_{□}(X) by definition of 𝕋_{□}(X) and 𝔉(m, T_{1}, T_{2}) = [U, ρ] by definition of 𝔉. The latter gives 𝔣(T_{1}) = [U, ρ], so 𝔣̄(T_{1}) = [U] = 𝔣̄(T) and consequently T_{1} ∈ {T, Ť}, so (2) holds.

Conversely, suppose that (2) holds and pick an element T_{1} of {T, Ť}∩𝕋_{□}(X). Then, by definition of 𝕋_{□}(X), we have (m, T_{1}, T_{2}) ∈ 𝕋_{0}(X) for some (m, T_{2}) ∈ ℤ^{+} × T_{∗}; and from the definition of 𝕋_{0}(X) it follows that (m, T_{1}, T_{2}) = disc(X, Λ, F) for some basic affine ruling Λ of X and some element F of Λ_{∗}. Then 5.3 and the definitions of 𝔣 and 𝔉 give

𝔣(T_{1}) = 𝔉(m, T_{1}, T_{2}) = res[X, Λ, F] = [X \ supp(F), ρ]

for some ρ : X \supp(F) → 𝔸^{1}; consequently 𝔣̄(T_{1}) = [X \supp(F)]. Since T_{1} ∈ {T, Ť}, 6.7 gives 𝔣̄(T_{1}) = 𝔣̄(T) = [U], so U ≅ X \ supp(F). □

Let a_{0}, a_{1}, a_{2} be pairwise relatively prime positive integers and consider ℙ = ℙ(a_{0}, a_{1}, a_{2}). In view of 7.5, the problem of deciding which members of ℳ_{0} can be embedded in ℙ reduces to describing the set 𝕋_{□}(ℙ). Now section 7 of [6] gives an explicit description of 𝕋_{0}(ℙ), and it is easy to derive a description of 𝕋_{□}(ℙ). See 8.4.

8. Examples

8.1. Given a surface U ∈ ℳ_{0}, find a tableau T ∈ T_{∗} such that 𝔣̄(T) = [U]. 8.1.1. Example. Let U = ℙ^{2} \C, where C ⊂ ℙ^{2} is “Yoshihara’s quintic” ([14], Prop. 3, cusp case). We show that U ∈ ℳ_{0} and we find T ∈ T_{∗} such that 𝔣̄(T) = [U].

From the equation of the curve, one finds that C has a unique singular point P_{0} (a cusp) and that if P_{1}, P_{2}, … are the points of C i.n. P_{0} then the multiplicity sequence {μ(P_{i}, C)}_{i=0}^{∞} is (2, 2, 2, 2, 2, 2, 1, 1, ...). It follows from the genus formula that C is rational, so U is (rational and) completable by rational curves. A sequence of 8 blowings-up (at P_{0}, …, P_{7}) gives U as the complement of an SNC-divisor D of a smooth projective surface S, where G(D, S) is:

[figure: −2; −1; −3; −2; −2; −2; −2; −2; −1]

It is easily verified that the above weighted graph is equivalent to the following chain of type (Z):

(79)  [figure: 0; −1; −5; −2; −2; −2; −2; −2]

so we get ML(U) = k by 2.20; in fact we have U ∈ ℳ_{0}, because Pic(U) = ℤ/5ℤ is a finite group. We seek T ∈ T_{∗} such that 𝔣̄(T) = [U]. We have the two graphs:

(80)  [figure: 0; −1; −5; −2; −2; −2; −2; −2; ∈ G_{∞}[U] and G_{res}[U] = ø; _{6}; _{25}]

where ø is the empty graph and the numbers under the braces are the determinants of the indicated subtrees. Compare (80) with (55) and (56), in the proof of 5.3. Since the determinant of the empty graph is 1, which is not equal to 25, we have T ∈ T_{2} by the last paragraph of the proof of 5.3. Writing T = ( \begin{matrix}p & 1 \\ c & a \end{matrix}) and comparing with (56), we find a = 1, ac^{2} = 25 and pc + 1 = 6; thus T = ( \begin{matrix}1 & 1 \\ 5 & 1 \end{matrix} ).

Note that (79) is equivalent to

(81)  [figure: 0; −1; −2; −2; −2; −2; −2; −5]

by 2.17, so (81) is another chain of type (Z) belonging to G_{∞}[U]. Making the above argument using (81) in place of (79) gives a different tableau, namely, ( \begin{matrix}4 & 1 \\ 5 & 1 \end{matrix} ). There is no contradiction because this is Ť and 𝔣̄(Ť) = 𝔣̄(T) by 6.7.

Note that the number of orbits is two, in Theorem C, because T ≠ Ť.

Finally, 7.1 implies that U is isomorphic to the complement of the curve X_{0}X_{2}+X_{1}^{5} = 0 in ℙ(1, 1, 4). 8.2. Given a tableau T ∈ T_{∗}, describe the surface U ∈ ℳ_{0} determined (up to isomorphism) by 𝔣̄(T) = [U]. 8.2.1. Example. Let T = ( \begin{matrix}1 & 1 \\ 3 & 2 \end{matrix} ) ∈ T_{∗}; let us describe the surface [U] = 𝔣̄(T).

By 7.1, U is isomorphic to the complement of the curve X_{0}X_{2} + X_{1}^{3} = 0 in ℙ(1, 2, 5). By 6.9, we may determine G_{res}[U] and an element of G_{∞}[U] by computing G_{(−1)} [figure] T

and G_{(−1)} [figure] T. The quick way to do this is to use (56) (because T ∈ T_{2}):

(82)  …  …^{0}  ^{−1}  _{acp+1=7} _{ac^{2}=18}  ^{−2}  ^{−2}  ^{−2}  ^{−2} \underbrace{G = underlying weighted graph of G_{(−1)} [figure] T}  \underbrace{G_{(−1)} [figure] T} _{a=2}

where we used T = ( \begin{matrix}p & 1 \\ c & a \end{matrix}) = ( \begin{matrix}1 & 1 \\ 3 & 2 \end{matrix} ). Since the determinant of the right-hand side is 2, there is only one vertex in G_{res}[U] = G_{(−1)} [figure] T. Let G denote the underlying weighted graph of the weighted pair G_{(−1)} [figure] T; then we know that G ∈ G_{∞}[U]. The subgraph of G obtained by deleting the two leftmost vertices is an admissible chain which is completely determined by the determinants 18 and 7, as explained in [6]-1.14. We find:

(83)  [figure: 0; −1; −3; −3; −2; −2; −2; \underbrace{G ∈ G_{∞}[U]}; \underbrace{G_{res}[U]}]

We may continue the description of U by noting that Pic(U_{s}) = ℤ/6ℤ (because μ(T) = 6). By looking at the graph G in (83), we see that U is not symmetric at infinity (this also follows from T ≠ Ť); so the number of orbits is 2, in Theorem C. 8.3. Example. Let us find all surfaces U ∈ ℳ_{0} such that Pic(U_{s}) has order 6. We use part (1) of 6.9. First, it is obvious that the elements of { T ∈ T_{∗} | μ(T) = 6 } are:

T_{1} = \binom{1}{6}, T_{2} = \binom{5}{6},

T_{3} = (\begin{matrix}1 & 1 \\ 6 & 1 \end{matrix}) , T_{4} = (\begin{matrix}5 & 1 \\ 6 & 1 \end{matrix}) , T_{5} = (\begin{matrix}1 & 1 \\ 3 & 2 \end{matrix}) , T_{6} = (\begin{matrix}2 & 1 \\ 3 & 2 \end{matrix}) , T_{7} = (\begin{matrix}1 & 1 \\ 2 & 3 \end{matrix}) .

Since T_{4} = Ť_{3} and T_{6} = Ť_{5}, we get five (non-isomorphic) surfaces, say U_{1}, U_{2}, U_{3}, U_{5}, U_{7}, where [U_{i}] = 𝔣̄(T_{i}). Of these, U_{3} is the only smooth one (6.9). As was shown in 8.2.1, each surface U_{i} can be described explicitely.

The last example describes 𝕋_{□}(ℙ) for ℙ = ℙ(1, 1, 1) = ℙ^{2}. This is derived from the description of 𝕋_{0}(ℙ^{2}) given in Example 7.8 of [6]. 8.4. Example. Define sequences {ξ_{n}}_{n=0}^{∞} and {u_{n}}_{n=0}^{∞} by:

ξ_{n} = 3ξ_{n−1} − ξ_{n−2}, ξ_{0} = −4, ξ_{1} = −1;
u_{n} = 3u_{n−1} − u_{n−2}, u_{0} = 1, u_{1} = 1.

Let {T_{n}}_{n=1}^{∞} be the sequence in T_{∗} defined by T_{1} = 1 (the empty tableau) and:

T_{n} = (\begin{matrix}ξ_{n} & 1 \\ u_{n} & 1 \end{matrix}) for n ≥ 2.

Explicitely:

\{T_{n}\}_{n=1}^{∞} = \{1, (\begin{matrix}1 & 1 \\ 2 & 1 \end{matrix}) , (\begin{matrix}4 & 1 \\ 5 & 1 \end{matrix}) , (\begin{matrix}11 & 1 \\ 13 & 1 \end{matrix}) , (\begin{matrix}29 & 1 \\ 34 & 1 \end{matrix}) , (\begin{matrix}76 & 1 \\ 89 & 1 \end{matrix}) , … \} .

Now it follows from [6]-7.8 that 𝕋_{□}(ℙ^{2}) = ⋃_{n=1}^{∞}{T_{n}, Ť_{n}}. Consequently,

(84) 𝔣̄(T_{1}), 𝔣̄(T_{2}), 𝔣̄(T_{3}), … are the members of ℳ_{0} which can be embedded in ℙ^{2}.

Comparing the list (84) with assertion (4) of 6.9, we see that many smooth surfaces belonging to ℳ_{0} cannot be embedded in ℙ^{2}. Since μ(T_{n}) = u_{n} holds for all n ≥ 1, the complement of 𝔣̄(T_{n}) in ℙ^{2} is a curve of degree u_{n}, so:

(85)

There exists a special curve of ℙ^{2} of degree d if and only if d is a term of {u_{n}}_{n=0}^{∞} = {1, 2, 5, 13, 34, 89, … }.

Moreover, the following fact can be derived from [6], but we don’t give the proof here:

(86)

If two special curves of ℙ^{2} have the same degree then some automorphism of ℙ^{2} maps one onto the other.

Remark. The special curve of ℙ^{2} of degree 5 is Yoshihara’s quintic (8.1.1), so the equation of a special curve can be much more complicated than X_{0}X_{1} + X_{2}^{n} = 0. However, one can show that if “f = 0” is the equation of a special curve then the hypersurface “f = 1” in 𝔸^{3} is isomorphic to a hypersurface with equation “xy = φ(z)”. Note that there may not exist an automorphism of 𝔸^{3} which maps one hypersurface onto the other. In other words, “f = 1” is a Danielewski surface which may be embedded in 𝔸^{3} in a non-standard way—the embedding is non-standard in the case of the quintic, and in fact in most cases.

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

E-mail address: ddaigle@uottawa.ca

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

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