AFFINE RULINGS OF WEIGHTED PROJECTIVE PLANES

DANIEL DAIGLE

Abstract. It is explained that the following two problems are equivalent: (i) Describing all affine rulings of any given weighted projective plane; (ii) describing all weighted-homogeneous locally nilpotent derivations of k[X, Y, Z]. Then the solution of (i) is outlined. (Outline of our joint work with Peter Russell.)

An “affine ruling” of an algebraic surface X is a morphism p : U → Γ where Γ is a curve, U is a nonempty open subset of X isomorphic to Γ × 𝔸^{1} and p is the projection Γ × 𝔸^{1} → Γ (note that we will modify this definition in Section 1). Then, how can we find all affine rulings on a given surface X? In our joint work with Peter Russell ([2] and [3]), we give a complete answer to that question in the case where X is a weighted projective plane, and a partial answer when X belongs to a larger class of surfaces.

The aim of this paper is to give an outline of [2] and [3] which is readable by a wider set of algebraists and geometers. To achieve this, we explain several notions which are well-known to geometers familiar with algebraic surfaces (but we do assume some familiarity with the notion of linear system on a surface); we also omit all proofs and present the material in an order which is quite different from that of [2] and [3].

Two distinct approaches are proposed in [2] and [3] but only the one via discrete data (section 5 of [2] and most of [3]) is outlined here. Although the other approach (via “X-immersions”) is necessary for a full understanding of the subject, it can be omitted in this type of outline. Also note that [2] and [3] contain several nontrivial results in the theory of weighted graphs, but that none of these appears here.

Motivation

There are several reasons for studying affine rulings, but let us explain the connection with locally nilpotent derivations. Consider the polynomial ring A = k[X_{0}, X_{1}, X_{2}], where k is an algebraically closed field of characteristic zero. It is known that describing the locally nilpotent derivations D : A → A is equivalent to describing their kernels; by a result of Miyanishi [7], the kernel of such a derivation is a subalgebra k[f, g] of A, where f and g are algebraically independent over k. So describing the locally nilpotent derivations D : A → A is equivalent to answering: Which pairs of polynomials f, g ∈ A have the property that k[f, g] is the kernel of a locally nilpotent derivation of A? However, this question seems to be very difficult. If we restrict ourselves to the case where D is (or equivalently f and g are) homogeneous with respect to weights w(X_{i}) = a_{i}, where a_{0}, a_{1}, a_{2} are relatively prime positive integers, then we can think of the zero sets of f and g as curves in the weighted projective plane ℙ(a_{0}, a_{1}, a_{2}) = Proj A; then [1] gives the following result: Theorem. For w-homogeneous elements f, g ∈ A satisfying gcd(w(f), w(g)) = 1, the following are equivalent:

2000 Mathematics Subject Classification. Primary: 14R10. Secondary: 14J26, 14R20, 13N15.

Key words and phrases. Algebraic surfaces, affine-ruled surfaces, weighted projective planes, locally nilpotent derivations, additive group actions on affine 3-space.

Research supported by NSERC Canada.

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

(2) there exists a w-homogeneous locally nilpotent derivation D of A such that ker D = k[f, g];

(3) f and g are irreducible elements of A and the complement of the set {fg = 0} in the weighted projective plane ℙ(a_{0}, a_{1}, a_{2}) is isomorphic to 𝔸_{∗}^{1} × 𝔸^{1} as an algebraic surface.

(Here, 𝔸_{∗}^{1} denotes the affine line minus one point.) Note that the case where gcd(w(f), w(g)) ≠ 1 turns out to be very special, and is completely described in [1]. Hence, describing homogeneous locally nilpotent derivations of A is equivalent to finding all pairs of curves C_{1}, C_{2} on ℙ(a_{0}, a_{1}, a_{2}) with the property that ℙ(a_{0}, a_{1}, a_{2})\(C_{1} ∪ C_{2}) is isomorphic to 𝔸_{∗}^{1} × 𝔸^{1}. Although this is not entirely obvious, it turns out that this is equivalent to finding all affine rulings of ℙ(a_{0}, a_{1}, a_{2}) (see Problems 1 and 2 of 1.3). So,

Describing the affine rulings of ℙ(a_{0}, a_{1}, a_{2}) is equivalent to describing the w-homo-geneous locally nilpotent derivations of A.

The class of homogeneous locally nilpotent derivations of A is not well understood, and corresponds to an interesting class of G_{a}-actions on 𝔸^{3}. In particular, our description of the affine rulings of ℙ(a_{0}, a_{1}, a_{2}) might eventually enable one to produce families of automorphisms of 𝔸^{3} which are currently unknown.

Conventions and terminology

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

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

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

(4) If Λ is a linear system on a surface X, then by a member of Λ we mean a divisor D of X such that D ∈ Λ. Assume that D has irreducible support, i.e., that D = nC for some n ∈ ℤ^{+} and some irreducible curve C; we call D a reduced member of Λ if n = 1, a multiple member if n > 1.

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

(6) Every graph considered in this paper is a finite undirected graph such that no edge connects a vertex to itself and at most one edge joins any given pair of vertices. (The words “graph” and “tree” are always used in this restricted sense.)

(7) A weighted graph is a graph in which each vertex is assigned an integer (called its weight).

(8) Let S be a smooth projective surface and D a divisor of S with strong normal crossings; let D_{1}, …, D_{n} be the distinct components of D. The dual graph of (D, S) is the weighted graph G = G(D, S) whose vertices are the components of D; distinct vertices D_{i} and D_{j} are joined by an edge if the curves D_{i} and D_{j} have nonempty intersection; and the weight of a vertex D_{i} is the self-intersection number D_{i}^{2} of the curve D_{i}. We say that D_{j} is a neighbor of D_{i} if i ≠ j and D_{i} ∩ D_{j} ≠ ∅ (i.e., if the vertices D_{i}, D_{j} of G are neighbors); the number of neighbors of D_{i} is called its branching number; if this number is strictly greater than 2, we say that D_{i} is a branching component of D (or that the vertex D_{i} is a branch point of G). We say that G is a linear chain (or a linear tree) if it is a tree without branch points; an admissible chain is a linear chain in which every weight is strictly less than −1; note that the empty graph is an admissible chain. We say that D is a tree (or a linear chain, or an admissible chain, etc) if G has the corresponding property.

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

For instance, if X^{∗} → X is a blowing-up morphism, then this process gives the strict transform Λ^{∗} of Λ. On the other hand, if X → X^{∗} is a blowing-up then the Λ^{∗} so obtained will simply be called the “image” of Λ (then Λ is the strict transform of Λ^{∗}).

(10) We will need to consider the Nagata ruled surface 𝔽_{m} for positive values of m; Λ_{m} denotes the standard ruling of 𝔽_{m} and Σ_{m} the negative section of Λ_{m} (Σ_{m} is a curve on 𝔽_{m} isomorphic to a projective line, Σ_{m}^{2} = −m and Σ_{m} · D = 1 for every D ∈ Λ_{m}).

1. Preliminaries

Definition of affine ruling

1.1. Let X be a projective normal rational surface. We claim that every “affine ruling” p : U → Γ of X (as defined in the introduction) determines a linear system Λ on X. Indeed, U ≅ Γ×𝔸^{1} is normal and rational, so Γ is an open subset of ℙ^{1}. The morphism p extends to a rational map X → ℙ^{1} which, in turn, determines a unique linear system Λ on X without fixed components.

It is proved in Section 1 of [2] that any two affine rulings of X determining the same linear system Λ differ in a trivial way. Since our task is to enumerate all affine rulings, we should not distinguish between rulings which determine the same linear system; so we adopt the viewpoint that Λ itself is the affine ruling: Definition. Let Λ be a one-dimensional linear system on X without fixed components. We say that Λ is an affine ruling of X if there exist nonempty open subsets U ⊂ X and Γ ⊆ ℙ^{1} such that U ≅ Γ × 𝔸^{1} and such that the projection morphism Γ × 𝔸^{1} → Γ determines Λ.

From now-on, “affine ruling” is always understood as in the above definition.

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

• the general member of Λ is irreducible and reduced;

• Λ has at most one base point on X.

A class of surfaces and three problems

1.2. The symbol X always denotes a projective algebraic surface which is at least normal and rational. Following the terminology of [2] and [3], we say that an algebraic surface X satisfies (‡) if:

(‡)

X is a projective normal rational surface, X is affine-ruled, rank(Pic X_{s}) = 1 and every singular point of X is a cyclic quotient singularity,

where X_{s} denotes the smooth locus of X. (Remarks: (i) “X is affine-ruled” means that there exists at least one affine ruling of X; (ii) the group Pic(X_{s}) is the same as the divisor class group of X.)

If a, b, c are any positive integers, the weighted projective plane ℙ(a, b, c) satisfies (‡). In particular, ℙ^{2} satisfies (‡). In Section 1 of [2], it is proved that if X satisfies (‡) then X has at most three singular points. We also know that the class (‡) contains many surfaces other than the weighted projective planes, and we have a characterization (see 1.6) of weighted projective planes in the class (‡). 1.3. Given an algebraic surface X satisfying (‡), consider: Problem 1. Find all affine rulings of X.

Problem 2. Find all pairs of curves C_{1}, C_{2} on X such that X \(C_{1} ∪C_{2}) is isomorphic to 𝔸_{∗}^{1} × 𝔸^{1}. Problem 3. Find all curves C in X such that κ̄(X_{s} \ C) = −∞.

It is shown in [2] that Problems 1 and 2 are equivalent, and that a solution to Problem 1 contains, in particular, a solution to Problem 3; also, some references are given in [2] for Problem 3 in the case X = ℙ^{2}.

Our aim is to investigate Problem 1 for an arbitrary X satisfying (‡), and in particular for X = ℙ(a_{0}, a_{1}, a_{2}).

Resolution graph of a surface

1.4. Definition. If X is a projective normal rational surface, then it makes sense to consider the minimal resolution of singularities ρ : X̂ → X of X. Then ρ^{−1}(Sing X) is the support of a divisor Ê of X̂ with strong normal crossings and, moreover, each connected component of Ê is a tree of projective lines. The dual graph G(Ê, X̂) is called the resolution graph of the surface X.

We recall a fact concerning resolution graphs: 1.5. For X as in 1.4, the following are equivalent:

(1) Every singular point of X is a cyclic quotient singularity;

(2) every connected component of the resolution graph of X is an admissible chain. Remark. As far as this article is concerned, the reader unfamiliar with the notion of “cyclic quotient singularities” may use 1.5 as a definition.

The following result was obtained in [3] after essentially everything else had been proved: 1.6. Theorem. Let X be a complete normal rational surface which is affine-ruled and satisfies rank(Pic X_{s}) = 1. If X has the same resolution graph as the weighted projective plane ℙ(a, b, c), then X is isomorphic to ℙ(a, b, c).

In other words, 1.6 characterizes weighted projective planes among all surfaces satisfying (‡).

Resolution graph of a weighted projective plane

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 A,

where A = k[X_{0}, X_{1}, X_{2}] is graded by assigning weight a_{i} to X_{i}. By 1.3.1 of [4], there exist pairwise relatively prime positive integers a′_{0}, a′_{1}, a′_{2} such that ℙ(a_{0}, a_{1}, a_{2}) ≅ ℙ(a′_{0}, a′_{1}, a′_{2}). From now-on, whenever a weighted projective plane ℙ(a_{0}, a_{1}, a_{2}) is under consideration, we will always assume that

a_{0}, a_{1}, a_{2} are pairwise relatively prime.

Let ℙ = ℙ(a_{0}, a_{1}, a_{2}) and consider the points q_{0} = (1 : 0 : 0), q_{1} = (0 : 1 : 0), q_{2} = (0 : 0 : 1) ∈ ℙ. Then it is known that Sing(ℙ) ⊆ {q_{0}, q_{1}, q_{2}}, where q_{i} is singular if and only if a_{i} > 1. Moreover, these are well understood cyclic quotient singularities, and the resolution graphs of such singularities were described in [5]. Consequently, the resolution graph of ℙ is known. We now proceed to describe it (see Section 1 of [3] for details). 1.7. (1) Given an ordered triple (a, b, c) of pairwise relatively prime positive inte-

gers, we define an admissible chain A_{(a,b,c)} as follows. Write r_{0} = a and let r_{1} be the unique integer satisfying 0 ≤ r_{1} < r_{0} and br_{1} ≡ c (mod a); consider the “outer” Euclidean algorithm on (r_{0}, r_{1}): r_{i−1} = q_{i}r_{i} − r_{i+1} (0 ≤ r_{i+1} < r_{i}, i = 1, …, n) and r_{n+1} = 0; then A_{(a,b,c)} is:

[figure: …; −q_{1}; −q_{2}; −q_{n−1}; −q_{n}].

Note that A_{(a,b,c)} is the empty chain if and only if a = 1. Also, one can see that if b and c are interchanged then the only difference is that the integers (q_{1}, …, q_{n}) are produced in the reverse order; so A_{(a,b,c)} = A_{(a,c,b)}.

(2) Given an unordered triple [a_{0}, a_{1}, a_{2}] of pairwise relatively prime positive inte-

gers, let G_{[a_{0},a_{1},a_{2}]} be the disjoint union of A_{(a_{0},a_{1},a_{2})}, A_{(a_{1},a_{2},a_{0})} and A_{(a_{2},a_{0},a_{1})}. 1.8. Proposition. Given pairwise relatively prime a_{0}, a_{1}, a_{2} ∈ ℤ^{+}, the resolution graph of ℙ(a_{0}, a_{1}, a_{2}) is G_{[a_{0},a_{1},a_{2}]}. Remark. In [3], any surface X satisfying (‡) and with resolution graph G_{[a_{0},a_{1},a_{2}]} is called a surface “of type [a_{0}, a_{1}, a_{2}]”. Then, near the end of the paper, it is shown that such a surface must be isomorphic to ℙ(a_{0}, a_{1}, a_{2}) (this is 1.6 in the present work). 1.9. Example. To find the resolution graph of ℙ(5, 6, 7) by using Definition 1.4, one has to minimally resolve the singularities and look at the resolution locus, i.e., the inverse image of the singular points (see Figure 1).

Figure 1. Resolution of singularities of ℙ(5, 6, 7)

[figure: −2; −2; −3; −2; −2; −2; −2; −4; −2]  ⟶ ℙ(5, 6, 7)

8 DANIEL DAIGLE

Then the resolution graph of ℙ(5, 6, 7) is the dual graph of the resolution locus:

[figure: −2; −4; −2; −2; −2; −2; −2; −3; −2]

However, it is quicker to use 1.8, which tells us that the resolution graph is G_{[5,6,7]}. Note that G_{[5,6,7]} is exactly the above graph.

2. Blowing-up according to a tableau

Blowing-up, blowing-down and equivalence of weighted graphs

2.1. Let G be a weighted graph. We define three types of “blowing-up of G”:

(1) If v is a vertex of G then the blowing-up of G at v is the weighted graph G′ obtained from G by adding one vertex e of weight −1, adding one edge joining e to v, and decreasing the weight of v by 1. (This process is called a blowing-up “at a vertex”, or a “sprouting” blowing-up.)

(2) If ϵ = {v_{1}, v_{2}} is an edge of G (so v_{1}, v_{2} are distinct vertices of G joined by an edge), then the blowing-up of G at ϵ is the weighted graph G′ obtained from G by adding one vertex e of weight −1, deleting the edge ϵ = {v_{1}, v_{2}}, adding the two edges {v_{1}, e} and {e, v_{2}}, and decreasing the weights of v_{1} and v_{2} by 1. (This is called a blowing-up “at an edge”, or a “subdivisional” blowing-up.)

(3) The free blowing-up of G is the weighted graph G′ obtained by taking the disjoint

union of G and of a vertex e of weight −1. We will use the symbol G ← G′ to indicate that G and G′ are weighted graphs and that G′ is a blowing-up of G. In each of the above three cases, we call e the vertex created by G ← G′. In reverse, we say that G is obtained by contracting (or blowing-down) G′ at e. 2.2. More precisely, given a weighted graph G′ and a vertex e of G′, the blowing-down of G′ at e is allowed if and only if the following three conditions hold: (i) e has weight −1; (ii) e has at most two neighbors in G′; (iii) if v_{1} and v_{2} are distinct neighbors of e in G′ then v_{1}, v_{2} are not neighbors in G′.

Given a sequence G_{0} ← ⋯ ← G_{n} of blowings-up, we may also speak of the contraction (or blowing-down) “ G_{n} ≥ G_{0} ” of weighted graphs. 2.3. Two weighted graphs are equivalent if one can be obtained from the other by a finite sequence of blowings-up and blowings-down. 2.4. Let S be a smooth projective surface and D a divisor of S with strong normal crossings. If S′ → S is the blowing-up of S at a point P then the inverse image in S′ of {P } ∪ Supp(D) is the support of a divisor D′ of S′ with strong normal crossings. Then the dual graph G(D′, S′) is a blowing-up of G(D, S).

Weighted pairs

2.5. By a weighted pair, we mean an ordered pair (G, v) where G is a (nonempty) weighted graph and v is a vertex of G. If (G, v) is a weighted pair, we call v its distinguished vertex. 2.6. Let (G, v) and (G′, v′) be weighted pairs. Suppose that G′ is a blowing-up of G (i.e., G ← G′) and that the following hold: (i) The blowing-up G ← G′ is either at v or at an edge incident to v; and (ii) v′ is the vertex of G′ which is created by the blowing-up G ← G′. Then we say that (G′, v′) is a blowing-up of (G, v) and write (G, v) ← (G′, v′).

2.7. 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 . 2.8. Let (G_{0}, e_{0}) be a weighted pair and \binom{p}{c} ∈ T a one column tableau. By blowing-up (G_{0}, e_{0}) according to \binom{p}{c}, we mean producing the sequence (G_{0}, e_{0}) ← ⋯ ← (G_{n}, e_{n}) defined as follows.

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

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

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

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

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

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

Remark. The sequence (G_{0}, e_{0}) ← ⋯ ← (G_{n}, e_{n}) of 2.8 “follows” the euclidean algorithm of (p, c) and satisfies:

(i) n ≥ 1 and equality holds if and only if \binom{p}{c} = \binom{1}{1}.

(ii) G_{0} ← G_{1} is the only blowing-up in (G_{0}, e_{0}) ← ⋯ ← (G_{n}, e_{n}) which is “at a

vertex”. 2.9. Let (G_{0}, e_{0}) be a weighted pair and T = ( \begin{matrix}p_{1} & ··· & p_{k} \\ c_{1} & ··· & c_{k} \end{matrix}) ∈ T a tableau.

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

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

• If k = 1, then (G_{0}, e_{0}) ← ⋯ ← (G_{n}, e_{n}) is defined in 2.8.

• If k > 1, then (G_{0}, e_{0}) ← ⋯ ← (G_{n}, e_{n}) is

(G_{0}, e_{0}) ← ⋯ ← (G_{m}, e_{m}) ← (G_{m+1}, e_{m+1}) ← ⋯ ← (G_{n}, e_{n}),

where (G_{0}, e_{0}) ← ⋯ ← (G_{m}, e_{m}) is the sequence obtained by blowing-up (G_{0}, e_{0}) according to \binom{p_{1}}{c_{1}} and (G_{m}, e_{m}) ← ⋯ ← (G_{n}, e_{n}) is obtained by blowing-up

(G_{m}, e_{m}) according to ( \begin{matrix}p_{2} & ··· & p_{k} \\ c_{2} & ··· & c_{k} \end{matrix}).

2.10. Given a weighted pair (G_{0}, e_{0}), consider the set S_{(G_{0},e_{0})} of sequences of blowingsup of weighted pairs (G_{0}, e_{0}) ← ⋯ ← (G_{n}, e_{n}) satisfying: if n > 0 then G_{0} ← G_{1} is the blowing-up of G_{0} at e_{0}.

The above paragraphs show that each T ∈ T determines an element of S_{(G_{0},e_{0})} (obtained by blowing-up (G_{0}, e_{0}) according to T). In fact, it is not difficult to see that this set map T → S_{(G_{0},e_{0})} is bijective. 2.11. Consider a pair (π, G), where π : S′ → S is a birational morphism of smooth projective surfaces and G ⊂ S is a smooth curve. Assume:

(1) G ∩ center(π) contains at most one point; (2) if G ∩ center(π) = {P }, then π^{−1}(P) has exactly one (−1)-component. Let us explain how (π, G) determines a tableau \overline{HN}(π, G) ∈ T . If G ∩ center(π) is empty, let \overline{HN}(π, G) be the empty tableau. If G ∩ center(π) = {P }, let n be the number of irreducible components of π^{−1}(P)

and factor π as

S = S_{0} ⟵^{π_{1}} ⋯ ⟵^{π_{n}} S_{n} ⟵^{π′} S′,

where π_{i} is the blowing-up of S_{i−1} at a point Q_{i} ∈ S_{i−1} infinitely near P (so Q_{1} = P), and where the center of π′ is disjoint from (π_{1} ⋯ π_{n})^{−1}(P) ⊂ S_{n}. Let E_{i} = π_{i}^{−1}(Q_{i}) ⊂ S_{i} (for 1 ≤ i ≤ n) and let E_{0} = G. Then E_{n} is a (−1)-component of π^{−1}(P) and, by assumption (2), it is the only one. It follows:

(3) P_{i} ∈ E_{i−1} for all i = 1, …, n, so we have a sequence of blowings-up of weighted pairs

(G(D_{0}, S_{0}), E_{0}) ← ⋯ ← (G(D_{n}, S_{n}), E_{n}),

where D_{i} is the inverse image of G in S_{i} (so D_{0} = G = E_{0}). By 2.10, there is a unique tableau T ∈ T such that this sequence is the blowing-up of (G(D_{0}, S_{0}), E_{0}) according to T. This tableau T is denoted \overline{HN}(π, G). Remark. \overline{HN}(π, G) is a simplified version of the Hamburger-Noether tableau ([8], [6]). Note that the Hamburger-Noether tableau contains more information than \overline{HN}(π, G).

3. Construction of affine rulings

This section gives a method for constructing all pairs (X, Λ) where X is a surface satisfying (‡) and Λ is an affine ruling of X. See the introduction for the notations 𝔽_{m}, Λ_{m} and Σ_{m}. 3.1. Definition. Fix a triple (m, T_{1}, T_{2}) ∈ ℤ^{+} × T × T .

(1) By a blowing-up of 𝔽_{m} according to (T_{1}, T_{2}), we mean a triple (π, P_{1}, P_{2}) where (a) π : Y → 𝔽_{m} is a birational morphism (with Y smooth and projective);

(b) P_{1}, P_{2} are points of 𝔽_{m} \ Σ_{m} belonging to distinct members of Λ_{m} (P_{i} ∈ Z_{i} ∈ Λ_{m}, Z_{1} ≠ Z_{2});

(c) center(π) ⊆ {P_{1}, P_{2}} and, for each i = 1, 2, π^{−1}(P_{i}) contains at most one (−1)-curve (so \overline{HN}(π, Z_{i}) is defined);

(d) \overline{HN}(π, Z_{i}) = T_{i} for i = 1, 2. Warning. The blowings-up of 𝔽_{m} according to (T_{1}, T_{2}) always exist, but are not unique.

(2) Let β = (π, P_{1}, P_{2}) be a blowing-up of 𝔽_{m} according to (T_{1}, T_{2}), with notation as in part 1. We define a divisor D_{β} of Y (with strong normal crossings) as follows. For each i = 1, 2, let

E_{i} = \{ \begin{matrix}strict transform of Z_{i} in Y, & if T_{i} = 1; \\ the (−1)-component of π^{−1}(P_{i}), & if T_{i} ≠ 1. \end{matrix}

Then let D_{β} be the reduced effective divisor of Y whose support is

π^{−1}(Z_{1} ∪ Σ_{m} ∪ Z_{2}) minus E_{1} and E_{2}.

It is easy to see that the dual graph G(D_{β}, Y) depends only on the discrete data (m, T_{1}, T_{2}), i.e., is independent of the choice of β. This weighted graph is denoted G(m, T_{1}, T_{2}).

Remark. In [2] and [3], the notation (G_{(−m)} ⊖T_{1})⊖T_{2} was used in place of G(m, T_{1}, T_{2}), but using the same notation here would force us to define many concepts. Concretely, the graph G(m, T_{1}, T_{2}) can be computed as follows:

(1) Let G be the weighted graph [figure: 0; z_{1}; −m; 0; z_{2}] (dual graph of (Z_{1} + Σ_{m} + Z_{2}, 𝔽_{m})).

(2) Let (G, z_{1}) ← ⋯ ← (G′, e_{1}) be the blowing-up of (G, z_{1}) according to T_{1}, and note that z_{2} is a vertex of G′.

(3) Let (G′, z_{2}) ← ⋯ ← (G′′, e_{2}) be the blowing-up of (G′, z_{2}) according to T_{2}; then

G(m, T_{1}, T_{2}) is obtained from G′′ by removing e_{1} and e_{2}. 3.2. Let 𝕋(‡) be the set of triples (m, T_{1}, T_{2}) ∈ ℤ^{+} × T × T such that:

(1) T_{1} satisfies one of the following three conditions:

(a) T_{1} = 1 (the empty tableau);

(b) T_{1} = \binom{p}{c}, for some \binom{p}{c} ≠ \binom{1}{1};

(c) T_{1} = ( \begin{matrix}p & 1 \\ c & N \end{matrix} ), for some \binom{p}{c} ≠ \binom{1}{1} and N ≥ 1.

(2) If T_{2} is nonempty then its first column is not \binom{1}{1}.

(3) Each connected component of the weighted graph G(m, T_{1}, T_{2}) shrinks to an

admissible chain. 3.3. We also define a subset 𝕋_{0}(‡) of 𝕋(‡) by:

𝕋_{0}(‡) = {(m, T_{1}, T_{2}) ∈ 𝕋(‡) | T_{2} also satisfies one of (a), (b), (c) in part 1 of 3.2} .

3.4. It is important to note that the sets 𝕋(‡) and 𝕋_{0}(‡) can be described explicitely. See “Some explicit computations” at the end of [2].

Given (m, T_{1}, T_{2}) ∈ 𝕋(‡) and a blowing-up β of 𝔽_{m} according to (T_{1}, T_{2}), the next result defines a pair (X_{β}, Λ_{β}) where X_{β} is a surface satisfying (‡) and Λ_{β} is an affine ruling of X_{β}. The notation (X_{β}, Λ_{β}) will always be used in this sense, i.e., the pair determined by β as in 3.5. 3.5. Theorem. Let τ = (m, T_{1}, T_{2}) ∈ 𝕋(‡), let β = (π, P_{1}, P_{2}) be any blowing-up of 𝔽_{m} according to (T_{1}, T_{2}) (with notation π : Y → 𝔽_{m}) and consider the divisor D_{β} of Y. Then the following hold:

(1) There exists a birational morphism σ : Y → X_{β} whose exceptional locus is exactly the support of D_{β}, and where X_{β} is a surface satisfying (‡).

(2) If Λ_{β} is the linear system on X_{β} which is determined by Λ_{m} via π and σ (i.e., Λ_{β} is the image under σ of the strict transform of Λ_{m}), then Λ_{β} is an affine ruling of X_{β}.

(3) Up to isomorphism, X_{β} is completely determined by τ.

3.6. Theorem. Let X be a surface satisfying (‡) and Λ an affine ruling of X. Then there exist (m, T_{1}, T_{2}) ∈ 𝕋(‡) and a blowing-up β of 𝔽_{m} according to (T_{1}, T_{2}) such that X is isomorphic to X_{β} and, under that isomorphism, Λ corresponds to Λ_{β}.

The above two theorems show how to construct all pairs (X, Λ) where X is a surface satisfying (‡) and Λ is an affine ruling of X. Since the set 𝕋(‡) can be described explicitely (3.4), this method is quite satisfactory. The next section will address the question of constructing all Λ on a given X.

An example

For the remainder of this section, we consider the surface X = ℙ(5, 6, 7) and a certain affine ruling Λ of X. Our aim is to illustrate how 3.6 can be proved, so we start from (X, Λ) and seek (m, T_{1}, T_{2}) and β.

We stress that Λ is a specific affine ruling of ℙ(5, 6, 7), i.e., we could (but we won’t) give explicit equations for it. The paragraphs below state several properties of Λ without justification, some of which are specific to this particular Λ (e.g. Figure 3). In other words, we claim that there exists an affine ruling Λ of ℙ(5, 6, 7) having all the properties described in the following paragraphs.

It is proved in [2] that if X satisfies (‡) and Λ is an affine ruling of X then every member of Λ has irreducible support and Λ has at most two multiple members. In our case, Λ has exactly two multiple members, of the form 5C_{1} and 21C_{2} where C_{1} and C_{2} are irreducible curves (see Figure 2). The other dotted curves in Figure 2 represent some of the reduced members of Λ.

Figure 2. X = ℙ(5, 6, 7), with affine ruling Λ

[figure: (0 : 1 : 0); base point of Λ; (0 : 0 : 1); (1 : 0 : 0); C_{2}; C_{1}]

We begin by “resolving” (X, Λ), i.e., by constructing a pair (X̃, Λ̃) = (X, Λ)^{∼} as follows:

(1) Minimally resolve the singularities of X (write X̂ → X). Let Λ̂ be the strict transform of Λ on X̂.

(2) Minimally resolve the base point of Λ̂ (write X̃ → X̂). Let Λ̃ be the strict

transform of Λ̂ on X̃. Let ρ : X̃ → X be the composition X̃ → X̂ → X. The center of ρ is Sing X ∪Bs Λ and ρ^{−1}(Sing X ∪Bs Λ) is the support of a divisor D of X̃ with strong normal crossings. For the pair (X, Λ) under consideration, the resolution (X̃, Λ̃) = (X, Λ)^{∼} is as in Figure 3.

Figure 3. Resolution of (X, Λ)

[figure: X̃; −1; H; −2; −3; −2; −2; −3; −2; −4; −2; −2; −4; −3; −2; C̃_{1}; C̃_{2}]

→

[figure: X̂; −2; −2; −3; −2; Ĉ_{1}; Ĉ_{2}; base point of Λ̂; −2; −2; −2; −4; −2]
↓
X = ℙ(5, 6, 7)

In Figure 3, note the following:

(1) In X̃ (resp. X̂), the divisor displayed in full lines is D (resp. the inverse image of Sing(X)) and the dotted curves are the strict transforms of the dotted curves in Figure 2; in particular, C̃_{i} (resp. Ĉ_{i}) is the strict transform of C_{i}.

(2) Since Λ̃ is base-point-free and has ℙ^{1} as general member, Λ̃ is a ℙ^{1}-ruling of X̃.

(3) The curve H ⊂ X̃ is the “last exceptional curve”, i.e., if we factor X̃ → X̂ as

X̃ = Y_{4} ⟶^{μ_{4}} Y_{3} ⟶^{μ_{3}} Y_{2} ⟶^{μ_{2}} Y_{1} ⟶^{μ_{1}} Y_{0} = X̂ (four monoidal transformations),

then H is the exceptional curve of μ_{4}. Consequently, H is a section of Λ̃ (i.e., we have H · M = 1 for every M ∈ Λ̃).

Next we claim:

(i) Λ̃ has exactly two reducible members, which we denote F_{1}, F_{2} ∈ Λ̃; (ii) each reducible member F_{i} has exactly one (−1)-component, namely, C̃_{i}; (iii) D = (F_{1}^{#} − C̃_{1}) + H + (F_{2}^{#} − C̃_{2}) where F_{i}^{#} is the reduced effective divisor of

X̃ with same support as F_{i}. (See Figure 4.)

Figure 4. Section and reducible members of Λ̃

[figure: (X̃, Λ̃); −1; H; −2; −3; F_{2}^{∘}; −2; −2; F_{1}^{∘}; −3; −2; −4; −2; −2; −4; −3; −2; .; .; .; .; .; .; .; .; .; .; C̃_{2}; −1; .; .; .; .; .; .; .; .; .; .; C̃_{1}; −1]

Note that, in Figure 4, if we consider only the full lines then we have a picture of D (the same picture as in Figure 3). But if we consider the full lines and also the two dotted lines C̃_{i}, the resulting divisor is D + C̃_{1} + C̃_{2}, which is equal to H + F_{1}^{#} + F_{2}^{#}.

It is a well-known fact that, given a ℙ^{1}-ruling on a smooth projective rational surface, the reducible members may be shrunk to 0-curves and this produces one of the Nagata ruled surfaces together with its standard ruling. Applying this to (X̃, Λ̃) gives a morphism π : X̃ → 𝔽_{1} whose exceptional locus is the support of (F_{1}^{#} − F_{1}^{∘}) + (F_{2}^{#} − F_{2}^{∘}), where F_{i}^{∘} is the unique irreducible component of F_{i} which meets H. Figure 5 shows the codomain of π.

In Figure 5, note the following:

(1) π(H) = Σ_{1}, π(F_{i}) = π(F_{i}^{∘}) = Z_{i} ∈ Λ_{1} (for i = 1, 2) and center(π) = {P_{1}, P_{2}}.

(2) For each i = 1, 2, π^{−1}(P_{i}) has a unique (−1)-component (namely, C̃_{i}). By the second observation, it makes sense to define T_{i} = \overline{HN}(π, Z_{i}) (see 2.11). In fact we have T_{1} = ( \begin{matrix}3 \\ 5 \end{matrix} ) and T_{2} = ( \begin{matrix}1 & 3 \\ 3 & 7 \end{matrix} ).

Now it is clear that β = (π, P_{1}, P_{2}) is a blowing-up of 𝔽_{1} according to (T_{1}, T_{2}) and that D_{β} = D. Consequently, the weighted graph G (1, ( \begin{matrix}3 \\ 5 \end{matrix} ) , ( \begin{matrix}1 & 3 \\ 3 & 7 \end{matrix} )) (defined in Figure 5. Codomain of π

[figure: 𝔽_{1} with its ruling Λ_{1}; −1; Σ_{1}; 0; P_{2}; Z_{2}; 0; P_{1}; Z_{1}]

3.1) is the dual graph of (D, X̃), which shrinks to the resolution graph of X; thus each connected component of G (1, ( \begin{matrix}3 \\ 5 \end{matrix} ) , ( \begin{matrix}1 & 3 \\ 3 & 7 \end{matrix} )) shrinks to an admissible chain and it follows from definition 3.2 that (m, T_{1}, T_{2}) = (1, ( \begin{matrix}3 \\ 5 \end{matrix} ) , ( \begin{matrix}1 & 3 \\ 3 & 7 \end{matrix} )) ∈ 𝕋(‡). Finally, note that (X_{β}, Λ_{β}) = (X, Λ).

4. Affine rulings of a given surface

4.1. Given a surface X satisfying (‡), we define a subset 𝕋(X) of 𝕋(‡) by

𝕋(X) = {(m, T_{1}, T_{2}) ∈ 𝕋(‡) | (m, T_{1}, T_{2}) satisfies conditions (1), (2)} ,

where (1) and (2) are the following equivalent conditions:

(1) For some blowing-up β of 𝔽_{m} according to (T_{1}, T_{2}), X_{β} ≅ X.

(2) For all blowings-up β of 𝔽_{m} according to (T_{1}, T_{2}), X_{β} ≅ X. That (1) and (2) are equivalent is a consequence of part 3 of 3.5, which also implies that if 𝕋(X) ∩ 𝕋(X′) ≠ ∅ then X ≅ X′ and 𝕋(X) = 𝕋(X′) (so the sets 𝕋(X) form a partition of 𝕋(‡)).

We also define the subset 𝕋_{0}(X) = 𝕋_{0}(‡) ∩ 𝕋(X) of 𝕋(X).

The following fact is proved in [2]: 4.2. For a surface X satisfying (‡), the following problems are equivalent:

(1) Finding all affine rulings of X (i.e., solving Problem 1);

(2) describing the set 𝕋(X);

(3) describing the set 𝕋_{0}(X).

In fact, 3.5 and 3.6 show that the first problem reduces to the second: Assume that the set 𝕋(X) is known and, for each (m, T_{1}, T_{2}) ∈ 𝕋(X), consider all blowings-up β of 𝔽_{m} according to (T_{1}, T_{2}); then the pairs (X_{β}, Λ_{β}) give all affine rulings of X.

The second problem reduces to the third because [2] gives a method for generating 𝕋(X) from its subset 𝕋_{0}(X) (see part 2 of 4.11, below).

For a general X satisfying (‡), we don’t know how to describe 𝕋(X) or 𝕋_{0}(X); however, thanks to 1.6, we can do it if X is a weighted projective plane. The idea is as follows.

4.3. Let X be a surface satisfying (‡) and let R denote the resolution graph of X. Given an arbitrary τ = (m, T_{1}, T_{2}) ∈ 𝕋(‡), consider the weighted graph G(τ) = G(m, T_{1}, T_{2}) defined as in 3.1. It is not difficult to see that the following conditions are equivalent:

(1) G(τ) ∼ R (equivalence of weighted graphs);

(2) for every blowing-up β of 𝔽_{m} according to (T_{1}, T_{2}), the surface X_{β} has resolution

graph R. By 1.6, if X is a weighted projective plane then the second condition is equivalent to X_{β} ≅ X, and hence to τ ∈ 𝕋(X). In other words,

𝕋(X) = {τ ∈ 𝕋(‡) | G(τ) ∼ R} and 𝕋_{0}(X) = {τ ∈ 𝕋_{0}(‡) | G(τ) ∼ R}

hold whenever X is a weighted projective plane. Thus the determination of 𝕋_{0}(ℙ(a, b, c)) reduces to solving: Problem G. Given pairwise relatively prime a, b, c ∈ ℤ^{+}, find all τ ∈ 𝕋_{0}(‡) such that G(τ) ∼ G_{[a,b,c]}.

Note that Problem G belongs to the theory of weighted graphs and makes no reference to geometry; it is solved in [3] using graph theory only. We will now state the solution.

Description of 𝕋_{0}(ℙ(a, b, c))

Given pairwise relatively prime positive integers a_{0}, a_{1}, a_{2}, paragraphs 4.4–4.7 define sets

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}).

Then 𝕋_{0}(ℙ(a, b, c)) is described in 4.8. Remark. From now on, the 2×1 matrix \binom{0}{1} should be interpreted as the empty tableau 1 ∈ T . 4.4. Given pairwise relatively prime positive integers a_{0}, a_{1}, a_{2}, it is clear that

Eq(a_{0}, a_{1}, a_{2}): a_{0} = a_{1}a_{2}x_{0} − a_{2}x_{1} − a_{1}x_{2} has a unique solution (x_{0}, x_{1}, x_{2}) ∈ ℕ^{3} satisfying 0 ≤ x_{1} < a_{1} and 0 ≤ x_{2} < a_{2}. Then x_{0} > 0 and for i = 1, 2 we have x_{i} = 0 ⟺ a_{i} = 1 and x_{i} ∈ {0, 1} ⟺ a_{i} | (a_{0} + a_{1} + a_{2}). For each i = 1, 2, there is a unique x′_{i} satisfying x_{i}x′_{i} ≡ 1 (mod a_{i}) and 0 ≤ x′_{i} < a_{i}; and a unique x′′_{i} ∈ ℤ satisfying x_{i}x′_{i} − x′′_{i}a_{i} = 1. 4.5. Let a_{0}, a_{1}, a_{2} be pairwise relatively prime positive integers.

(1) The set 𝕋_{I}(a_{0}, a_{1}, a_{2}) has exactly one element, namely

(x_{0}, \binom{x_{1}}{a_{1}}, \binom{x_{2}}{a_{2}}) ,

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

(2) The set 𝕋_{II.1}(a_{0}, a_{1}, a_{2}) has at most one element, and is nonempty if and only if

\frac{a_{0} + a_{1} + a_{2}}{a_{2}} is a natural number strictly greater than 2. Moreover, if 𝕋_{II.1}(a_{0}, a_{1}, a_{2}) is nonempty then let (x_{0}, x_{1}, x_{2}) be the unique solution to Eq(a_{0}, a_{1}, a_{2}), let

x′_{1}, x′′_{1} be as in 4.4 and define

\binom{p_{2}}{c_{2}} = \binom{a_{1} − x_{1} − x′_{1} + x′′_{1}}{a_{1} − x_{1}} + (x_{0} − x_{2})\binom{a_{1} − x′_{1}}{a_{1}};

then the unique element of 𝕋_{II.1}(a_{0}, a_{1}, a_{2}) is

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

(3) 𝕋_{II.2}(a_{0}, a_{1}, a_{2}) = {(m, T_{2}, T_{1}) | (m, T_{1}, T_{2}) ∈ 𝕋_{II.1}(a_{0}, a_{1}, a_{2})}. 4.6. Let a_{0}, a_{1}, a_{2} be pairwise relatively prime positive integers satisfying a_{1}a_{2} | a_{0} + a_{1} + a_{2}, and write γ = \frac{a_{0}+a_{1}+a_{2}}{a_{1}a_{2}} . Then (a_{0}, a_{1}, a_{2}) determines two sets, W_{(a_{0},a_{1},a_{2})} and W^{(a_{0},a_{1},a_{2})}, which we now proceed to define.

(1) Each 2 × 2 matrix M (with entries in ℤ) determines a pair of sequences s(M) = (s_{0}, s_{1}, s_{2}, …), t(M) = (t_{0}, t_{1}, t_{2}, …)

defined by (\begin{matrix}s_{0} & s_{1} \\ t_{0} & t_{1} \end{matrix}) = M and \begin{cases}s_{n−1} + s_{n+1} = a_{2}γt_{n} \\ t_{n−1} + t_{n+1} = a_{1}γs_{n} \end{cases}.

(2) Let M = ( \begin{matrix}1 & 1 \\ 1 & 1 \end{matrix} ) and define u_{n} = s_{n}(M) and v_{n} = t_{n}(M).

(3) Let M′ = ( \begin{matrix}−γ−1+x_{2} & x_{1}−1 \\ −γ−1+x_{1} & x_{2}−1 \end{matrix} ), where (x_{0}, x_{1}, x_{2}) is the solution to Eq(a_{0}, a_{1}, a_{2}),

and define ξ_{n} = s_{n}(M′) and η_{n} = t_{n}(M′).

(4) For every n ∈ ℕ, define

f_{n} = (1, ( \begin{matrix}ξ_{n} & 1 \\ u_{n} & a_{1} \end{matrix} ) , ( \begin{matrix}v_{n+1}−η_{n+1} & 1 \\ v_{n+1} & a_{2} \end{matrix} )) and g_{n} = (1, ( \begin{matrix}u_{n+1}−ξ_{n+1} & 1 \\ u_{n+1} & a_{1} \end{matrix} ) , ( \begin{matrix}η_{n} & 1 \\ v_{n} & a_{2} \end{matrix} )) .

Then define

W_{(a_{0},a_{1},a_{2})} = \{ \begin{matrix}\{f_{2}, g_{3}, f_{4}, g_{5}, … \}, & if a_{0} > a_{1} − a_{2}, \\ ∅, & else, \end{matrix}

and

W^{(a_{0},a_{1},a_{2})} = \{ \begin{matrix}\{g_{2}, f_{3}, g_{4}, f_{5}, … \}, & if a_{0} > a_{2} − a_{1}, \\ ∅, & else. \end{matrix}

4.7. Let a_{0}, a_{1}, a_{2} be pairwise relatively prime positive integers. Then 𝕋_{III}(a_{0}, a_{1}, a_{2}) is nonempty if and only if a_{1}a_{2} | a_{0} + a_{1} + a_{2}, in which case we have:

𝕋_{III}(a_{0}, a_{1}, a_{2}) = W_{(a_{0},a_{1},a_{2})} ∪ W^{(a_{0},a_{1},a_{2})}.

Remark. If a_{1}a_{2} | a_{0} + a_{1} + a_{2} and a_{0} > |a_{1} − a_{2}|, then

𝕋_{III}(a_{0}, a_{1}, a_{2}) = {f_{2}, f_{3}, f_{4}, … } ∪ {g_{2}, g_{3}, g_{4}, … }.

Also observe that 𝕋_{III}(a_{0}, a_{2}, a_{1}) = {(m, T_{2}, T_{1}) | (m, T_{1}, T_{2}) ∈ 𝕋_{III}(a_{0}, a_{1}, a_{2})} holds in all cases.

4.8. Theorem. Let (a, b, c) be pairwise relatively prime positive integers. Then 𝕋_{0}(ℙ(a, b, c)) is the union of the sets

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}) ∪ 𝕋_{III}(a_{0}, a_{1}, a_{2})

for all permutations (a_{0}, a_{1}, a_{2}) of (a, b, c). 4.9. Example. The following is a description of 𝕋_{0}(ℙ^{2}) (recall that ℙ^{2} = ℙ(1, 1, 1)). First, 4.5 gives:

• 𝕋_{I}(1, 1, 1) = {(1, 1, 1)} (where 1 is the empty tableau);

• 𝕋_{II.1}(1, 1, 1) = \{(1, 1, ( \begin{matrix}1 & 1 \\ 2 & 1 \end{matrix} ))\};

• 𝕋_{II.2}(1, 1, 1) = \{(1, ( \begin{matrix}1 & 1 \\ 2 & 1 \end{matrix} ) , 1)\}.

We have 𝕋_{III}(1, 1, 1) = {f_{2}, f_{3}, f_{4}, … } ∪ {g_{2}, g_{3}, g_{4}, … } by 4.7; by 4.6 (with γ = 3), we find that u_{n} = v_{n} and ξ_{n} = η_{n} for all n, and:

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

So,

𝕋_{III}(1, 1, 1) = \{(1, ( \begin{matrix}1 & 1 \\ 2 & 1 \end{matrix} ) , ( \begin{matrix}1 & 1 \\ 5 & 1 \end{matrix} )) , (1, ( \begin{matrix}4 & 1 \\ 5 & 1 \end{matrix} ) , ( \begin{matrix}2 & 1 \\ 13 & 1 \end{matrix} )) , (1, ( \begin{matrix}11 & 1 \\ 13 & 1 \end{matrix} ) , ( \begin{matrix}5 & 1 \\ 34 & 1 \end{matrix} )) , … \}

∪ \{(1, ( \begin{matrix}1 & 1 \\ 5 & 1 \end{matrix} ) , ( \begin{matrix}1 & 1 \\ 2 & 1 \end{matrix} )) , (1, ( \begin{matrix}2 & 1 \\ 13 & 1 \end{matrix} ) , ( \begin{matrix}4 & 1 \\ 5 & 1 \end{matrix} )) , (1, ( \begin{matrix}5 & 1 \\ 34 & 1 \end{matrix} ) , ( \begin{matrix}11 & 1 \\ 13 & 1 \end{matrix} )) , … \}

and 𝕋_{0}(ℙ^{2}) is the union of the above four sets. 4.10. Example. We now describe 𝕋_{0}(ℙ(2, 3, 5)). By 4.5,

• 𝕋_{I}(2, 3, 5) = \{(1, \binom{2}{3}, \binom{1}{5})\};

• 𝕋_{I}(2, 5, 3) = \{(1, \binom{1}{5}, \binom{2}{3})\};

• 𝕋_{I}(3, 2, 5) = \{(1, \binom{1}{2}, \binom{1}{5})\};

• 𝕋_{I}(3, 5, 2) = \{(1, \binom{1}{5}, \binom{1}{2})\};

• 𝕋_{I}(5, 2, 3) = \{(2, \binom{1}{2}, \binom{2}{3})\};

• 𝕋_{I}(5, 3, 2) = \{(2, \binom{2}{3}, \binom{1}{2})\};

• 𝕋_{II.1}(3, 5, 2) = \{(1, \binom{1}{5}, ( \begin{matrix}3 & 1 \\ 4 & 2 \end{matrix} ))\};

• 𝕋_{II.2}(3, 2, 5) = \{(1, ( \begin{matrix}3 & 1 \\ 4 & 2 \end{matrix} ) , \binom{1}{5})\};

• 𝕋_{II.1}(5, 3, 2) = \{(1, \binom{2}{3}, ( \begin{matrix}1 & 1 \\ 4 & 2 \end{matrix} ))\};

• 𝕋_{II.2}(5, 2, 3) = \{(1, ( \begin{matrix}1 & 1 \\ 4 & 2 \end{matrix} ) , \binom{2}{3})\}.

We have 𝕋_{III}(3, 5, 2) = {g_{2}, f_{3}, g_{4}, f_{5}, … } by 4.7; by 4.6 (with γ = 1),

u_{n−2} + u_{n} = 2v_{n−1}, u_{0} = 1, u_{1} = 1;
v_{n−2} + v_{n} = 5u_{n−1}, v_{0} = 1, v_{1} = 1;
ξ_{n−2} + ξ_{n} = 2η_{n−1}, ξ_{0} = −1, ξ_{1} = 0;
η_{n−2} + η_{n} = 5ξ_{n−1}, η_{0} = −1, η_{1} = 0,

so 𝕋_{III}(3, 5, 2) = \{(1, ( \begin{matrix}5 & 1 \\ 7 & 5 \end{matrix} ) , ( \begin{matrix}1 & 1 \\ 4 & 2 \end{matrix} )) , (1, ( \begin{matrix}2 & 1 \\ 7 & 5 \end{matrix} ) , ( \begin{matrix}22 & 1 \\ 31 & 2 \end{matrix} )) , (1, ( \begin{matrix}39 & 1 \\ 55 & 5 \end{matrix} ) , ( \begin{matrix}9 & 1 \\ 31 & 2 \end{matrix} )) , … \}. Also,

𝕋_{III}(3, 2, 5) = {(m, T_{2}, T_{1}) | (m, T_{1}, T_{2}) ∈ 𝕋_{III}(3, 5, 2)}

= \{(1, ( \begin{matrix}1 & 1 \\ 4 & 2 \end{matrix} ) , ( \begin{matrix}5 & 1 \\ 7 & 5 \end{matrix} )) , (1, ( \begin{matrix}22 & 1 \\ 31 & 2 \end{matrix} ) , ( \begin{matrix}2 & 1 \\ 7 & 5 \end{matrix} )) , (1, ( \begin{matrix}9 & 1 \\ 31 & 2 \end{matrix} ) , ( \begin{matrix}39 & 1 \\ 55 & 5 \end{matrix} )) , … \} .

Then 𝕋_{0}(ℙ(2, 3, 5)) is the union of the above twelve sets. 4.11. Conclusion. Let ℙ = ℙ(a, b, c) where a, b, c are pairwise relatively prime positive integers. To describe the affine rulings of ℙ, one proceeds as follows.

(1) Using 4.4–4.8, describe 𝕋_{0}(ℙ) explicitely (4.9 and 4.10 are examples of this).

(2) Generate 𝕋(ℙ) from its subset 𝕋_{0}(ℙ). (Each element τ of 𝕋_{0}(ℙ) determines a set [τ, ∞) which can be described explicitely by using 5.39 of [2]; by 5.22 of [2], 𝕋(ℙ) is the union of the sets [τ, ∞) for τ ∈ 𝕋_{0}(ℙ).)

(3) For each τ = (m, T_{1}, T_{2}) ∈ 𝕋(ℙ), consider all blowings-up β of 𝔽_{m} according to (T_{1}, T_{2}) and all corresponding pairs (X_{β}, Λ_{β}) (defined by 3.5). Then X_{β} ≅ ℙ and the Λ_{β} are all affine rulings of ℙ.

References

1. D. Daigle, Homogeneous locally nilpotent derivations of k[x, y, z], J. Pure Appl. Algebra 128 (1998), 109–132.

2. D. Daigle and P. Russell, Affine rulings of normal rational surfaces, CICMA report (Concordia, Laval, McGill), 1998-09, 59 pages.

3. , On weighted projective planes and their affine rulings, CICMA report (Concordia, Laval, McGill), 1999-02, 43 pages.

4. I. Dolgachev, Weighted projective varieties, Group Actions and Vector Fields (Proceedings, Vancouver 1981), Lectures Notes in Math., vol. 956, Springer-Verlag, 1982, pp. 34–71.

5. F. Hirzebruch, Über vierdimensionale Riemannsche Flächen mehrdeutiger Funktionen von zwei komplexen Veränderlichen, Math. Ann. 126 (1953), 1–22.

6. M. Koras and P. Russell, C^{∗}-actions on C^{3}: The smooth locus of the quotient is not of hyperbolic type, CICMA report (Concordia, Laval, McGill), 1996, 93 pages.

7. M. Miyanishi, Normal affine subalgebras of a polynomial ring, Algebraic and Topological Theories— to the memory of Dr. Takehiko MIYATA, Kinokuniya, 1985, pp. 37–51.

8. K.P. Russell, Hamburger-Noether expansions and approximate roots of polynomials, Manuscripta Math. 31 (1980), 25–95.

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

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