ON WEIGHTED PROJECTIVE PLANES AND THEIR AFFINE RULINGS

DANIEL DAIGLE AND PETER RUSSELL

Following the terminology of [2], we say that an algebraic surface X satisfies (†) if:

(†)

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; we say that X satisfies (‡) if:

(‡)

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

(Here, and throughout this paper, all algebraic varieties are over an algebraically closed field k of characteristic zero.) As we will see in Section 1, the weighted projective planes ℙ(a, b, c) satisfy (‡).

Paper [2] investigates the problem of finding all affine rulings of a given surface X satisfying (†). In particular, it shows that if X satisfies (‡) then the problem reduces to that of describing a certain set 𝕋_{0}(X) of triples (m, T_{1}, T_{2}), where m is a positive integer and each T_{i} is a 2 × h_{i} matrix with entries in ℕ (0 ≤ h_{i} ≤ 2). The aim of the present paper is to give an explicit description of the set 𝕋_{0}(X) in the case where X is a weighted projective plane; this is achieved by Corollary 7.1 and Propositions 7.3, 7.4 and 7.7. Thus [2] and this paper solve the above mentioned problem for weighted projective planes.

Let us also point out the following characterization of weighted projective planes, which we prove in the form of Corollary 6.12, below (see 1.19 for the notion of resolution graph of a normal surface): 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).

Although this paper relies heavily on the results and concepts developed in [2], it is almost completely self-contained, thanks to Section 2, which is essentially an outline of those parts of [2] which are directly needed here. However, it may be necessary to consult [2] in order to fully understand how to recover the affine rulings from the description of 𝕋_{0}(X) given in this paper. (First, starting from 𝕋_{0}(X), one uses 5.17 and 5.39 of [2] to construct the larger set 𝕋(X); then, as explained in 5.3 of [2], one has a “recipe” for constructing all affine rulings of X.)

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

Research supported by CICMA and grants from NSERC Canada.

We also refer to the introduction of [2] for a discussion of related problems and applications. For instance, the results of this paper enable one to describe all curves C on ℙ = ℙ(a, b, c) satisfying κ̄(ℙ \ C) = −∞, and all locally nilpotent derivations of k[X, Y, Z] which are homogeneous with respect to weights a, b, c for X, Y, Z.

1. Preliminaries on weighted projective planes

Let a_{0}, a_{1}, a_{2} be positive integers and consider the weighted projective plane

ℙ = 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}. Note that ℙ is a complete normal rational surface and that ℙ(a_{0}, a_{1}, a_{2}) ≅ ℙ(ȧ_{0}, ȧ_{1}, ȧ_{2}), where ȧ_{i} = a_{i}/d, d = gcd(a_{0}, a_{1}, a_{2}). Moreover, if we assume that a_{0}, a_{1}, a_{2} are relatively prime then: 1.1 ([3], 1.3.1). For distinct i, j, k ∈ \{0, 1, 2\}, let α_{i} = gcd(a_{j}, a_{k}) and a′_{i} = \frac{a_{i}}{α_{j}α_{k}}. Then a′_{0}, a′_{1}, a′_{2} are pairwise relatively prime and ℙ(a_{0}, a_{1}, a_{2}) ≅ ℙ(a′_{0}, a′_{1}, a′_{2}).

Since our results depend only on the isomorphism type of ℙ, and not on a specific projective structure, we will assume throughout: 1.2. a_{0}, a_{1}, a_{2} are pairwise relatively prime. 1.3. By a coordinate system of ℙ, we mean an ordered triple (f_{0}, f_{1}, f_{2}) of homogeneous elements of A satisfying A = k[f_{0}, f_{1}, f_{2}]. (Then (a_{0}, a_{1}, a_{2}) = (deg f_{τ0}, deg f_{τ1}, deg f_{τ2}) for some permutation τ of 0, 1, 2, and X_{i} ↦ f_{τi} gives an automorphism of A as a graded k-algebra.)

If F ∈ A is homogeneous, let V (F) ⊂ ℙ denote the zero locus of F. 1.4. Given a coordinate system (X_{0}, X_{1}, X_{2}) of ℙ, let R_{i} = V (X_{i}) ⊂ ℙ (an irreducible rational curve) and let q_{i} ∈ ℙ be the point R_{j} ∩ R_{k} (where {i, j, k} = {0, 1, 2}). Lemma 1.5. Given a coordinate system (X_{0}, X_{1}, X_{2}) of ℙ, the rational maps φ_{i} : ℙ → ℙ^{1} (i = 0, 1, 2) defined by

φ_{0} = \frac{X_{1}^{a_{2}}}{X_{2}^{a_{1}}} , φ_{1} = \frac{X_{2}^{a_{0}}}{X_{0}^{a_{2}}} , φ_{2} = \frac{X_{0}^{a_{1}}}{X_{1}^{a_{0}}}

induce three affine rulings of ℙ.

Proof. Note that q_{0} is the only fundamental point of φ_{0} in ℙ. The general fibre of φ_{0} is C = V (αX_{1}^{a_{2}} − βX_{2}^{a_{1}}), α, β ∈ k^{∗}, which is irreducible since gcd(a_{1}, a_{2}) = 1. Since C \ {q_{0}} ≅ 𝔸^{1}, φ_{0} induces an affine ruling of ℙ. □

Definition 1.6. The three affine rulings of 1.5 are said to be standard with respect to (X_{0}, X_{1}, X_{2}). An affine ruling of ℙ is standard if it is standard with respect to some coordinate system. Lemma 1.7. Let ℙ_{s} be the smooth locus of ℙ. Then Pic ℙ_{s} = ℤ.

Proof. We have Pic(ℙ_{s}) = Cl(ℙ_{s}) = Cl(ℙ), where “Cl” denotes divisor class group. Using the fact that A is an ℕ-graded U.F.D., one obtains a degree function Cl(ℙ) → ℤ which is in fact an isomorphism. □

By the above results, ℙ satisfies (†); we will show in 1.20 that ℙ satisfies a condition stronger than (‡). Also recall: 1.8 ([2], 1.16). A surface satisfying (‡) cannot have more than 3 singular points.

Linear chains

1.9. We use the standard definitions for blowing-up, contraction and equivalence of weighted graphs (but note that, in weighted graphs, we do not allow multiple edges between a given pair of vertices). A linear chain is a weighted tree without branch points; an admissible chain is a linear chain in which every weight is strictly less than −1. The empty graph is regarded as an admissible chain. 1.10. Let G be a weighted graph, v_{1}, …, v_{n} its vertices and ω_{i} the weight of v_{i}. Recall that the determinant of G is defined by det(G) = det(−A), where A denotes the “intersection matrix” of G, i.e., the n × n matrix with entries A_{ii} = ω_{i} and, if i ≠ j, A_{ij} = 1 (resp. 0) if v_{i}, v_{j} are neighbors (resp. are not neighbors). 1.11. Let G be a weighted tree, v a vertex of weight Ω(v) in G, G_{1}, …, G_{n} the branches of G at v and v_{i} the vertex of G_{i} which is a neighbor of v in G. If d_{i} = det G_{i} and d′_{i} = det(G_{i} − {v_{i}}), then

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

Definition 1.12. Let A be the linear chain [figure: …; w_{1}; w_{2}; w_{n−1}; w_{n}; v_{1}; v_{2}; v_{n−1}; v_{n}] (w_{i} ∈ ℤ, n ≥ 0). We say that A has discriminant δ and subdiscriminants δ^{∗} and δ_{∗} to indicate that det(A) = δ and that {det(A \ {v_{1}}), det(A \ {v_{n}})} = {δ^{∗}, δ_{∗}} (equality of sets). If A is empty, it has discriminant 1 and subdiscriminants 0 and 0; if A consists of a single vertex, its subdiscriminants are 1 and 1. 1.13. If A is a linear chain with discriminant δ and subdiscriminants δ^{∗} and δ_{∗}, then δ^{∗}δ_{∗} ≡ 1 (mod δ). 1.14. Let A be an admissible chain with discriminant δ and let s be one of the subdiscriminants of A. Then 0 ≤ s < δ; also, A is empty ⟺ δ = 1 ⟺ s = 0. Moreover, A is completely determined by the outer Euclidian algorithm on (δ, s): write r_{0} = δ, r_{1} = s, 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 is

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

1.15. Let A and A′ be two linear chains.

(1) If A and A′ are equivalent as weighted graphs, then they have the same discriminant δ and, modulo δ, the same subdiscriminants.

(2) Assume that A and A′ are equivalent to admissible chains. If A and A′ have the same discriminant δ and if some subdiscriminants s of A and s′ of A′ satisfy s ≡ s′ (mod δ), then A and A′ are equivalent weighted graphs.

1.16. Let a, b, c be pairwise relatively prime positive integers.

(1) There is a unique integer c′ = c′(a, b) with 0 ≤ c′ < c and b ≡ ac′ mod c. (Note that c′ = 0 if and only if c = 1.)

(2) Define the integer c′′ = c′′(a, b) by c′(a, b)c′(b, a) = 1 + c′′c. (Note that c = 1 ⇒

c′′ = −1 and c ≠ 1 ⇒ 0 ≤ c′′ < c′ < c.) One also defines integers a′(b, c), a′(c, b), a′′(b, c), b′(a, c), etc. Note that each one of these is a function of the three variables a, b, c. Definition 1.17. Consider an unordered triple [δ_{0}, δ_{1}, δ_{2}], where δ_{0}, δ_{1}, δ_{2} are pairwise relatively prime positive integers. We define the weighted graph G_{[δ_{0},δ_{1},δ_{2}]} to be the disjoint union A_{0} ∪ A_{1} ∪ A_{2}, where A_{i} is the unique admissible chain with discriminant δ_{i} and subdiscriminants δ′_{i}(δ_{i+1}, δ_{i+2}) and δ′_{i}(δ_{i+2}, δ_{i+1}) (with indices computed modulo 3). Note that each A_{i} is allowed to be empty.

Cyclic quotient singularities

Let ω_{c} ⊂ k^{∗} be the group of c-th roots of unity. Lemma 1.18 ([4]). Let a, b, c be pairwise relatively prime positive integers. Let ω_{c} act on k[[ξ, η]] with weights a, b mod c for ξ, η and let X = Spec k[[ξ, η]]^{ω_{c}}.

(1) The exceptional locus of the minimal resolution of the singularity of X is an admissible chain E = E_{1} + ⋯ + E_{s} of rational curves with dual graph:

[figure: …; −q_{1}; −q_{2}; −q_{s−1}; −q_{s}; E_{1}; E_{2}; E_{s−1}; E_{s}; _{c′′(a,b)}; _{c′(a,b)}; _{c}; ^{c′(b,a)}]

where the braces give the determinants of the indicated subtrees.

(2) The proper transform of the image of V (η) (resp. V (ξ)) meets E normally in

E_{1} (resp. E_{s}). 1.19. The resolution graph of a normal surface X is the dual graph of E in X̂, where E is the exceptional locus of the minimal resolution of singularities π : X̂ → X of X. Let x be a cyclic quotient singularity of X and recall that the resolution locus π^{−1}(x) of x is an admissible chain A. We define the discriminant and subdiscriminants of the singularity x to be those of A. A smooth point is regarded as a cyclic quotient singularity of discriminant 1. If the singularity x is determined by ω_{c} acting with weights a and b (where a, b, c are pairwise relatively prime) then Lemma 1.18 says that x has discriminant c and subdiscriminants c′(a, b) and c′(b, a).

Singularities of ℙ

Choose a coordinate system (X_{0}, X_{1}, X_{2}) of ℙ and consider the open neighbourhood D_{+}(X_{2}) of q_{2} in ℙ. As noted in the proof of 1.3.3 of [3], D_{+}(X_{2}) is isomorphic to the quotient 𝔸^{2}/ω_{a_{2}}, where the action is given by t(u_{0}, u_{1}) = (t^{a_{0}}u_{0}, t^{a_{1}}u_{1}) (with t ∈ ω_{a_{2}}, (u_{0}, u_{1}) ∈ 𝔸^{2}). So q_{2} is a cyclic quotient singularity of ℙ and, by 1.19, q_{2} has discriminant a_{2} and subdiscriminants a′_{2}(a_{0}, a_{1}) and a′_{2}(a_{1}, a_{0}); note that the image in ℙ of the line “u_{i} = 0” is part of R_{i} (i = 0, 1). Similar remarks hold for q_{0} and q_{1}, so we obtain: 1.20. (1) For each i = 0, 1, 2, ℙ has a cyclic quotient singularity at q_{i}, of discrimi-

nant a_{i} and subdiscriminants a′_{i}(a_{i+1}, a_{i+2}) and a′_{i}(a_{i+2}, a_{i+1}).

(2) Sing ℙ ⊆ {q_{0}, q_{1}, q_{2}}.

(3) q_{i} is a smooth point if and only if a_{i} = 1.

It follows that ℙ is a surface of type [a_{0}, a_{1}, a_{2}], according to: Definition 1.21. Let [δ_{0}, δ_{1}, δ_{2}] be an unordered triple of pairwise relatively prime positive integers. By a surface of type [δ_{0}, δ_{1}, δ_{2}], we mean a surface satisfying (‡) and whose resolution graph is G_{[δ_{0},δ_{1},δ_{2}]}. Remark. A surface X satisfying (‡) may or may not have a type as defined in 1.21. If X has a type, we sometimes say that it has tuned singularities. Remark. We will show in 6.12 that every surface of type [δ_{0}, δ_{1}, δ_{2}] is isomorphic to ℙ(δ_{0}, δ_{1}, δ_{2}).

Let ℙ̂ → ℙ be the minimal resolution of singularities and Q_{i} the exceptional locus above q_{i}. By the above, R_{0} and R_{1} meet the chain Q_{2} normally at opposite ends. (With some abuse of notation, we use the same letter to denote R_{i} and its proper transform in ℙ̂.) More precisely, we have the first part of the following lemma. The second part will be proved in section 3 (but will not be needed). Lemma 1.22. (1) R = R_{0} + Q_{1} + R_{2} + Q_{0} + R_{1} + Q_{2} is a “ring” of rational curves

with dual graph

[figure: …; R_{0}; R_{1}; R_{2}; ^{Q_{0}}; _{a′_{0}(a_{1},a_{2})}; ; ; ; ; a′_{2}(a_{0},a_{1}); a′_{1}(a_{2},a_{0}); Q_{2}; Q_{1}]

(2) −R is a canonical divisor of ℙ̂.

2. Graphs, tableaux and rulings

This section gathers some of the definitions and results of [2] and (we hope) organizes them in a coherent way. It also includes a few items which are not found in [2].

Graphs and tableaux

2.1. Given weighted graphs G and G′, the symbol G ← G′ indicates that G′ is obtained from G by blowing-up once. In that case, if V (resp. V′) denotes the set of vertices of G (resp. G′) then V can be viewed as a subset of V′ and V′ \ V contains a single vertex, say e. We call e the vertex created by G ← G′. This e has weight −1 and has at most two neighbors in G′; if it has one neighbor v_{1} (resp. two neighbors v_{1}, v_{2}) then, regarding v_{1} (resp. v_{1}, v_{2}) as a vertex of G, we say that G ← G′ is the blowing-up of G at the vertex v_{1} (resp. at the edge {v_{1}, v_{2}}). A blowing-up at a vertex (resp. at an edge) is also called a sprouting (resp. subdivisional) blowing-up. In reverse, we say that G is obtained by contracting (or blowing-down) G′ at e. Given a sequence G_{0} ← ⋯ ← G_{n} of blowings-up, we may also speak of the contraction “ G_{n} ≥ G_{0} ” of weighted graphs. 2.2. Let n ≥ 1. By a weighted n-tuple, we mean an ordered n-tuple S = (G, v_{1}, …, v_{n−1}) where G is a weighted graph and v_{1}, …, v_{n−1} are distinct vertices of G.

When n = 1, S is simply a weighted graph; when n = 2, it is called a weighted pair. The following is the only weighted n-tuple with n > 2 that we will need: Notation 2.3. Given x ∈ ℤ, let G_{(x)} denote the weighted triple (G, v_{1}, v_{2}), where G is

the weighted graph [figure: 0; v_{1}; x; 0; v_{2}] . 2.4. If (G, v) is a weighted pair, we call v its distinguished vertex. By a linear weighted pair, we mean a weighted pair (G, v) satisfying: (i) G is a linear chain; and (ii) v has at most one neighbor in G. 2.5. Let (G, v) be a weighted pair and G ≥ G′ a contraction of weighted graphs such that v is not contracted (i.e., v is still a vertex of G′). Then we write (G, v) ≥ (G′, v) and call this a contraction of weighted pairs. The equivalence relation (on the set of weighted pairs) generated by ≥ is denoted “≈”, and is called “equivalence of weighted pairs”. 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′). Remark. A blowing-up of weighted pairs (G, v) ← (G′, v′) cannot be undone by contracting (G′, v′) as in 2.5. 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 . We define a binary operation on the set T by:

( \begin{matrix}p_{1} & ··· & p_{k} \\ c_{1} & ··· & c_{k} \end{matrix}) ( \begin{matrix}p_{k+1} & ··· & p_{ℓ} \\ c_{k+1} & ··· & c_{ℓ} \end{matrix} ) = ( \begin{matrix}p_{1} & ··· & p_{k} & p_{k+1} & ··· & p_{ℓ} \\ c_{1} & ··· & c_{k} & c_{k+1} & ··· & c_{ℓ} \end{matrix} ) .

Thus T is the free monoid on the set of columns \binom{p}{c} where p ≤ c are relatively prime positive integers. 2.8. Let (G_{0}, e_{0}) be a weighted pair and \binom{p}{c} ∈ T . 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}

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.

(1) 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) Consider the sequence (G_{0}, e_{0}) ← ⋯ ← (G_{n}, e_{n}) obtained by blowing-up (G_{0}, e_{0}) according to T, as defined in part (1). Then we write (G_{0}, e_{0})T = (G_{n}, e_{n}). Hence, blowing-up is a right action of the monoid T on the set of weighted pairs.

2.10. Let S be a weighted n-tuple, with n ≥ 2, and let T ∈ T be a tableau.

Write S = (G, v_{1}, …, v_{n−1}) and let (G′, e) denote the weighted pair (G, v_{1})T, as defined in part (2) of 2.9. Note that v_{2}, …, v_{n−1} can be regarded as vertices of G′ \{e}.

(1) Define ST = (G′, e, v_{2}, …, v_{n−1}), a weighted n-tuple.

(2) Define S ⊖ T = (G′ \ {e}, v_{2}, …, v_{n−1}), a weighted (n − 1)-tuple.

(3) Let S [figure] T denote the unique connected component of S ⊖ T which contains no vertex of G. We regard S [figure] T as a weighted graph; actually, S [figure] T is a (possibly empty) admissible chain. Note that S [figure] T is empty when T is the empty tableau.

(4) Let S [figure] T be the complement of S [figure] T in S⊖T. We regard S [figure] T as a weighted (n − 1)-tuple.

Note that S ⊖ T is the disjoint union of S [figure] T and S [figure] T. 2.11. Given relatively prime positive integers a and b, define \binom{a}{b}^{∗} = \binom{x}{y}, where x and y are the unique nonnegative integers which satisfy

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

2.12 ([2], 3.23). Let c > p > 0 be relatively prime integers, let G be the weighted graph which consists of a single vertex v of weight zero, and let (G′, v′) = (G, v)\binom{p}{c}. Then G′ has two branches at v′, with determinants of subtrees as follows:

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

where we define \binom{p′′}{p′} = \binom{p}{c}^{∗}. Note that these two branches are (G, v) [figure] \binom{p}{c} (left part of the picture) and (G, v) [figure] \binom{p}{c} (right).

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

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

(This connected component is the same thing as (G′, v′) [figure] \binom{1}{N} = (G, v) [figure] ( \begin{matrix}p & 1 \\ c & N \end{matrix} ).) 2.13. Consider a weighted pair

[figure: L :; …; 0; v; −1; ω_{1}; ω_{n}; _{r_{1}}; _{r_{0}}]

where v is the distinguished vertex, n ≥ 0, ω_{i} ≤ −2, and where r_{0} and r_{1} denote the determinants of the indicated subtrees (if n = 1 then r_{1} = 1; if n = 0 then r_{0} = 1 and r_{1} = 0). Then L determines the 2 × 2 matrix M(L) = ( \begin{matrix}x & r_{0}−r_{1} \\ y & r_{0} \end{matrix} ), where ( \begin{matrix}x \\ y \end{matrix}) = ( \begin{matrix}r_{0}−r_{1} \\ r_{0} \end{matrix} )^{∗}. For each ν ≥ 0, let \binom{p_{ν}}{c_{ν}} = M(L) · \binom{1}{ν} (matrix product). Then define a subset T (L) of

ON WEIGHTED PROJECTIVE PLANES AND THEIR AFFINE RULINGS 9

T by:

T (L) = \{ \binom{p_{ν}}{c_{ν}}\binom{1}{1}^{ν} | ν ≥ 0 (resp. ν > 0)\}

if ω_{i} < −2 for some i (resp. ω_{i} = −2 for all i), and where \binom{p_{ν}}{c_{ν}}\binom{1}{1}^{ν} is a product in the monoid T . We also define T_{k}(L) = \{T ∈ T | T \binom{1}{1}^{k} ∈ T (L)\} for each k ∈ ℕ.

2.14. Given L as in 2.13, define L^{t}: [figure: …; 0; −1; ω_{n}; ω_{1}]. Also define L^{t^{0}} = L and, for each s > 0, L^{t^{s}} = (L^{t^{s−1}})^{t}. By 3.24 of [2], M(L^{t}) = M(L)^{t}. 2.15 ([2], 3.32). Given L as in 2.13 and \binom{p}{c} ∈ T such that \binom{p}{c} ≠ \binom{1}{1},

L\binom{p}{c} contracts to a linear weighted pair ⟺ \binom{p}{c} ∈ T_{k}(L) for some k ∈ ℕ.

Moreover, if \binom{p}{c} ∈ T_{k}(L) then L\binom{p}{c}\binom{1}{1}^{k} ≈ L^{t}. 2.16. We will sometimes refer to the following conditions on a tableau T ∈ T :

(1) T = 1 (the empty tableau);

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

(3) T = ( \begin{matrix}p & 1 \\ c & N \end{matrix} ), where \binom{p}{c} ≠ \binom{1}{1} and N ≥ 1.

Given T ∈ T satisfying one of the above conditions (1–3), define Ť ∈ T by:^{1}

Ť = \{ \begin{matrix}1, & if T satisfies 2.16.1; \\ \binom{p′}{c}, & if T satisfies 2.16.2, where p′ is given by \binom{p′′}{p′} = \binom{p}{c}^{∗} (see 2.11); \\ ( \begin{matrix}c−p & 1 \\ c & N \end{matrix} ) , & if T satisfies 2.16.3. \end{matrix}

Note that if T satisfies condition 2.16.i (where i ∈ {1, 2, 3}) then so does Ť. If s is a positive integer, write T^{(ˇ s)} = (T^{(ˇ (s−1))})ˇ , where T^{(ˇ 0)} = T. Note that T^{(ˇ 2)} = T.

Let ℤ^{+} denote the set of positive integers. 2.17. Let 𝕋(‡) be the set of triples (m, T_{1}, T_{2}) ∈ ℤ^{+} × T × T such that (i) T_{1} satisfies one of the conditions (1–3) of 2.16; (ii) T_{2} ∉ \binom{1}{1}T (i.e., if T_{2} is nonempty then its first column is not \binom{1}{1}); and (iii) each connected component of the weighted graph (G_{(−m)} ⊖ T_{1}) ⊖ T_{2} shrinks to an admissible chain. 2.18. Define an order relation > on the set 𝕋(‡) by declaring that (n, T_{1}, T_{2}) > (m, T′_{1}, T′_{2}) if n = 1 and the following holds (let L = G_{(−1)} [figure] T′_{1}):

There exist an integer s ≥ 1 and tableaux X_{1}, …, X_{s} such that T_{1} = (T′_{1})^{(ˇ s)}, T_{2} = X_{s} ⋯ X_{1}T′_{2} and X_{i} ∈ T_{k_{i}}(L^{t^{i}}), where k_{1} = m − 1 and k_{i} = 0 for all i > 1.

2.19. Consider τ = (1, T_{1}, T_{2}) ∈ 𝕋(‡) and let L′ = G_{(−1)} [figure] T_{1}. Then the following are equivalent:

1In the second part of the definition of Ť, we could also define p′ by 0 < p′ < c and pp′ ≡ 1 (mod c).

(1) τ is non-minimal in 𝕋(‡);

(2) T_{2} is nonempty and its first column belongs to T_{k}(L′) for some k ∈ ℕ. 2.20. Given (n, T_{1}, T_{2}), (m, T′_{1}, T′_{2}) ∈ 𝕋(‡), write (n, T_{1}, T_{2}) ≡ (m, T′_{1}, T′_{2}) to indicate that (G_{(−n)} ⊖ T_{1})T_{2} ≈ (G_{(−m)} ⊖ T′_{1})T′_{2} (equivalence of weighted pairs). Note that “≡” is an equivalence relation on the set 𝕋(‡). We have

τ > τ′ ⟹ τ ≡ τ′ (all τ, τ′ ∈ 𝕋(‡))

by 5.18 of [2], but “≡” is not the equivalence relation generated by “>”.

Affine rulings

2.21. Let X be a complete normal rational surface. By an affine ruling of X we mean a one-dimensional linear system Λ on X (without fixed components) which arises^{2} from 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. 2.22. Let Λ be an affine ruling of a surface X satisfying (‡). By “resolving” (X, Λ), we mean constructing a pair (X̃, Λ̃) = (X, Λ)^{∼} as follows ([2], 1.5):

(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. 2.22.1 ([2], 1.14). We say that Λ is basic if each connected component of D is a linear chain.

Then Theorem 2.1 of [2] implies (in particular): 2.22.2. Every surface satisfying (‡) admits a basic affine ruling.

Clearly, Λ̃ is base-point-free and its general member is ℙ^{1}, i.e., Λ̃ is a “ℙ^{1}-ruling” of X̃. Using that X satisfies (‡), one shows ([2], 1.8 and 1.15):

(1) Exactly one irreducible component H of D is a section of Λ̃.

(2) Each reducible G ∈ Λ̃ has exactly one (−1)-component C_{G}. Moreover, H ·C_{G} = 0 and D = H + ∑_{i}(G_{i}^{#} − C_{G_{i}}), where the G_{i} are the reducible members of Λ̃ and where G_{i}^{#} is the reduced effective divisor of X̃ with same support as G_{i}.

(3) Λ̃ has at most two reducible members. Define m > 0 by H^{2} = −m and consider the Nagata ruled surface 𝔽_{m}; let Λ_{m} be the standard ruling of 𝔽_{m} and Σ_{m} the negative section of Λ_{m}. Then well-known properties of ℙ^{1}-rulings imply:

2Note that Γ must be an open subset of ℙ^{1}, so p extends to a rational map p′ : X → ℙ^{1} and p′ determines a linear system Λ on X without fixed components.

(4) By shrinking each G_{i} to a 0-curve, we get π : X̃ → 𝔽_{m}, where the exceptional locus of π is disjoint from H, π(H) = Σ_{m} and π(G_{i}) ∈ Λ_{m}.

It follows from (2) that each member of Λ is irreducible (but not necessarely reduced). Via the isomorphism X̃ \ supp(D) ≅ X \ (Sing X ∪ Bs Λ), each F ∈ Λ determines an F̃ ∈ Λ̃; moreover, F ↦ F̃ is a bijection Λ → Λ̃. 2.22.3 ([2], 2.4 and 2.5). Define a nonempty subset Λ_{∗} of Λ by declaring that it contains all F ∈ Λ satisfying: (i) At most one element of Λ \ {F } is not reduced; and (ii) all branching components of D are components of F̃.

Note that if P_{i} ∈ 𝔽_{m} is a point of the center of π then π^{−1}(P_{i}) contains exactly one (−1)-curve (namely, C_{G_{i}}). Because of this property, π can be described by using a pair of Hamburger-Noether tableaux (one for each point of the center), say HN_{1} and HN_{2}. Let T_{i} be the tableau obtained from HN_{i} by deleting the third row and dividing each column by its gcd (T_{i} = \overline{HN}_{i} ∈ T , see 3.6 of [2]). The triple (m, T_{1}, T_{2}) is then a partial description of π. 2.23 ([2], 5.1 and 5.2). Given a triple (X, Λ, F), where X is a surface satisfying (‡), Λ is an affine ruling of X and F is an element of Λ_{∗}, let us now define an element τ of 𝕋(‡), called the discrete part of (X, Λ, F) (notation: disc(X, Λ, F) = τ). Consider the triple (m, T_{1}, T_{2}) constructed at the end of 2.22, but make sure^{3} that the P_{i}’s and G_{i}’s have been labeled in such a way that the bijection Λ → Λ̃ sends F to G_{2}. Then we define disc(X, Λ, F) = (m, T_{1}, T_{2}). It satisfies:

(m, T_{1}, T_{2}) ∈ 𝕋(‡) and (G_{(−m)} ⊖ T_{1}) ⊖ T_{2} is the dual graph of D in X̃,

so (G_{(−m)} ⊖ T_{1}) ⊖ T_{2} shrinks to the resolution graph of X (D and X̃ are as in 2.22 and 𝕋(‡) was defined in 2.17). 2.24 ([2], 5.25). Two triples as in 2.23 are equivalent, (X, Λ, F) ∼ (X′, Λ′, F′), when there exists an isomorphism X → X′ which transforms Λ into Λ′ and F into F′. If this is the case then (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(‡) → 𝕋(‡) [X, Λ, F] ↦ discrete part of [X, Λ, F]

where 𝕊(‡) is the set of equivalence classes [X, Λ, F]. This map is in fact surjective and restricts to a bijection

disc : 𝕊_{0}(‡) → 𝕋_{0}(‡)

where

𝕋_{0}(‡) = {(m, T_{1}, T_{2}) ∈ 𝕋(‡) | T_{2} satisfies one of conditions (1–3) of 2.16}
𝕊_{0}(‡) = {[X, Λ, F] ∈ 𝕊(‡) | Λ is basic} = disc^{−1}(𝕋_{0}(‡)).

3This can always be arranged; it may involve choosing some of the P_{i}’s and G_{i}’s when Λ̃ has less than two reducible members. See [2] for details.

2.25. Given X satisfying (‡), define subsets 𝕋_{0}(X) ⊂ 𝕋(X) of 𝕋(‡) by:

𝕋(X) = {disc(X, Λ, F) | Λ is an affine ruling of X and F ∈ Λ_{∗}} ,
𝕋_{0}(X) = 𝕋(X) ∩ 𝕋_{0}(‡)

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

Then 5.13 of [2] implies: For any τ, τ′ ∈ 𝕋(‡) satisfying τ ≡ τ′, we have

(1)  τ ∈ 𝕋(X) ⟺ τ′ ∈ 𝕋(X).

Moreover, if τ = disc(X, Λ, F) then there exists an affine ruling Λ′ of X and an element F′ of Λ′_{∗} such that supp(F) = supp(F′) and τ′ = disc(X, Λ′, F′). Note that these facts still hold if we replace the assumption τ ≡ τ′ by τ > τ′ (see 2.20). We also point out that 5.17 of [2] implies:

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

2.26. Noting that each element (m, T_{1}, T_{2}) of 𝕋_{0}(‡) satisfies exactly one of: I: Each of T_{1}, T_{2} has at most one column; II.1: T_{1} has at most one column but T_{2} has two; II.2: T_{1} has two columns but T_{2} has at most one; III: each of T_{1}, T_{2} has two columns,

we give the following two definitions:

(1) Given P ∈ {I, II.1, II.2, III}^{4} and pairwise relatively prime positive integers a_{0}, a_{1}, a_{2}, let 𝕋_{P}(a_{0}, a_{1}, a_{2}) be the set

{(m, T_{1}, T_{2}) ∈ 𝕋_{0}(‡) | (m, T_{1}, T_{2}) satisfies P and G_{i} ∼ A_{i} for i = 0, 1, 2} ,

where “∼” is equivalence of weighted graphs, G_{0} = (G_{(−m)} [figure] T_{1}) [figure] T_{2}, G_{1} = G_{(−m)} [figure] T_{1}, G_{2} = (G_{(−m)} [figure] T_{1}) [figure] T_{2} and where A_{i} is the unique admissible chain with discriminant a_{i} and subdiscriminants a′_{i}(a_{i+1}, a_{i+2}) and a′_{i}(a_{i+2}, a_{i+1}) (with indices computed modulo 3). Note that G_{0}, G_{1} and G_{2} are the connected components of (G_{(−m)} ⊖ T_{1}) ⊖ T_{2}, with the understanding that G_{1} and G_{2} are allowed to be empty (G_{0} is never empty).

(2) Let Λ be a basic affine ruling of a surface X satisfying (‡). Then it is easy to see that

{disc(X, Λ, F) | F ∈ Λ_{∗}} = {(m, T_{1}, T_{2}), (m, T_{2}, T_{1})} ⊂ 𝕋_{0}(X)

for some tableaux T_{1} and T_{2} and some m ∈ ℤ^{+}. We say that Λ is a basic affine ruling of type I (resp. II, III ) if, for F ∈ Λ_{∗}, the discrete part (m, T_{1}, T_{2}) of (X, Λ, F) satisfies the above condition I (resp. II.1 or II.2, III).

4We mean that P is one of the four symbols I, II.1, II.2, III.

2.27. Let X be a surface of type [a, b, c], where a, b, c are pairwise relatively prime positive integers (see 1.21). If Λ is a basic affine ruling of X and F ∈ Λ_{∗} then τ = disc(X, Λ, F) belongs to 𝕋_{P}(a_{0}, a_{1}, a_{2}) for some P ∈ {I, II.1, II.2, III} and some permutation a_{0}, a_{1}, a_{2} of a, b, c. (Indeed, if we write τ = (m, T_{1}, T_{2}) then (G_{(−m)}⊖T_{1})⊖T_{2} is equivalent to the resolution graph of X, which is G_{[a,b,c]}.) 2.28. Let P ∈ {I, II.1, II.2, III} and let a_{0}, a_{1}, a_{2} be pairwise relatively prime positive integers. If (m, T_{1}, T_{2}) ∈ 𝕋_{P}(a_{0}, a_{1}, a_{2}) then the entry in the lower right corner of T_{i} is a_{i}. (For i ∈ \{1, 2\} we may write G_{i} = Z [figure] T_{i} = Z [figure] \binom{p_{i}}{c_{i}}, where G_{i} is as in 2.26, Z is the weighted pair consisting of a single vertex of weight zero and \binom{p_{i}}{c_{i}} is the rightmost column of T_{i}; then 2.12 gives det(G_{i}) = c_{i}, so c_{i} = a_{i}.)

Note that 2.28 holds even when T_{i} is empty, in which case we use the following convention: 2.29. When a tableau T has at most one column, we sometimes abuse notation and write T = \binom{p}{c} in all cases, with p = 0 and c = 1 when T is empty. 2.30. Suppose that τ = (m, T_{1}, T_{2}), τ′ = (m′, T′_{1}, T′_{2}) ∈ 𝕋(‡) satisfy τ ≡ τ′ and consider G_{0}, G_{1}, G_{2} determined by τ as in 2.26 and G′_{0}, G′_{1}, G′_{2} determined by τ′ in a similar way. Then it is immediate that G_{1} ∼ G′_{1} and that, for some permutation i, j of 0, 2, G_{0} ∼ G′_{i} and G_{2} ∼ G′_{j}. In the special case where τ > τ′, we have:

If T′_{2} is nonempty (resp. empty), then G_{i} ∼ G′_{i} (resp. G_{i} ∼ G′_{2−i}) for all i = 0, 1, 2.

If τ is a non-minimal element of 𝕋(‡) then ([2], 5.21) there exists a unique τ^{−} ∈ 𝕋(‡) satisfying: (i) τ > τ^{−} and (ii) no τ′ ∈ 𝕋(‡) is such that τ > τ′ > τ^{−}. We call τ^{−} the immediate predecessor of τ. Lemma 2.31. Let τ be a nonminimal element of 𝕋(‡), let τ^{−} be its immediate predecessor and suppose that τ ∈ 𝕋_{P}(a_{0}, a_{1}, a_{2}) for some P ∈ {I, II.1, II.2, III} and some pairwise relatively prime positive integers a_{0}, a_{1}, a_{2}. Then

τ^{−} ∈ \{ \begin{matrix}𝕋_{II.2}(a_{0}, a_{1}, a_{2}), & if P = III, \\ 𝕋_{I}(a_{0}, a_{1}, a_{2}), & if P = II.1, \\ 𝕋_{P}(a_{2}, a_{1}, a_{0}), & if P ∈ \{I, II.2\}. \end{matrix}

Proof. Write τ = (1, T_{1}, T_{2}) and τ^{−} = (m, T′_{1}, T′_{2}) and recall that T_{1} and T′_{1} have the same number of columns, and the number of columns of T′_{2} is strictly less than that of T_{2}. If P is I or II.2 then T′_{2} must be the empty tableau, so the assertion follows from 2.30.

Suppose that P = III (resp. P = II.1). If T′_{2} is not empty then, again, the assertion follows from 2.30. Assume that T′_{2} is empty and note that τ^{−} ∈ 𝕋_{II.2}(a_{2}, a_{1}, a_{0}) (resp. τ^{−} ∈ 𝕋_{I}(a_{2}, a_{1}, a_{0})) by 2.30. Since τ > τ^{−} and T′_{2} = 1, we have T_{2} ∈ T_{m−1}(L^{t}) by definition of “>” (where L = G_{(−1)} [figure] T′_{1}); since T_{2} has two columns, its rightmost column is therefore \binom{1}{1} and we get a_{2} = 1 by 2.28. Applying 2.28 to τ^{−} gives a_{0} = 1, so (a_{2}, a_{1}, a_{0}) = (a_{0}, a_{1}, a_{2}) and consequently τ^{−} ∈ 𝕋_{II.2}(a_{0}, a_{1}, a_{2}) (resp. τ^{−} ∈ 𝕋_{I}(a_{0}, a_{1}, a_{2})). □

3. Basic affine rulings of type I

The following uses the convention of 2.29:

Lemma 3.1. Let Λ_{0}, Λ_{1}, Λ_{2} be the standard affine rulings of ℙ = ℙ(a_{0}, a_{1}, a_{2}) with respect to a coordinate system (X_{0}, X_{1}, X_{2}) (where Λ_{i} corresponds to the φ_{i} of 1.5). Let i, j, k be a permutation of 0, 1, 2.

(1) For some F ∈ (Λ_{i})_{∗}, supp F = R_{j}.

(2) The discrete part of (ℙ, Λ_{i}, F) is (z, \binom{x}{a_{j}}, \binom{y}{a_{k}}), where (x, y, z) is the unique integral solution of a_{i} = a_{j}a_{k}z − a_{j}y − a_{k}x with 0 ≤ x < a_{j} and 0 ≤ y < a_{k}.

Proof. It’s enough to prove the case (i, j, k) = (0, 1, 2). Consider Λ = Λ_{0}. Clearly, there exist F_{1}, F_{2} ∈ Λ_{∗} such that supp F_{i} = R_{i}. Consider (ℙ̃, Λ̃) = (ℙ, Λ)^{∼} and the morphisms ℙ̃ → ℙ̂ → ℙ. Consider the divisor R of ℙ̂ as in Lemma 1.22.

Since Bs(Λ) = {q_{0}}, and since the strict transforms of R_{1}, R_{2} on ℙ̃ belong to distinct members of Λ̃, we have:

(i) If ℙ̃ → ℙ̂ is the identity map, then some component of Q_{0} is a section of Λ̂; (ii) if ℙ̃ → ℙ̂ is not the identity map, then it is centered at a point of Q_{0} and is

subdivisional for R − R_{0}.

Hence, the divisor H +∑_{i} G_{i}^{#} of ℙ̃ (notation as in 2.22) is a linear chain; it follows that Λ is basic of type I and that the discrete part of (ℙ, Λ, F_{1}) has the form (z, \binom{x}{c_{1}}, \binom{y}{c_{2}}), with 2.29 in effect. Moreover, the connected components of the weighted graph

(G_{(−z)} ⊖ \binom{x}{c_{1}}) ⊖ \binom{y}{c_{2}}

have determinants c_{1}, c_{2} and zc_{1}c_{2} −c_{1}y −c_{2}x, and are respectively equal to Q_{1}, Q_{2} and to a chain which contracts to Q_{0}. So c_{1} = a_{1}, c_{2} = a_{2} and a_{0} = za_{1}a_{2} − a_{1}y − a_{2}x. □

Proof of Lemma 1.22. Let the notation be as in the above proof; we show that −R is a canonical divisor of ℙ̂.

Consider the inverse image R̃ of R in ℙ̃; let π : ℙ̃ → 𝔽_{m} be the contraction of the reducible members of Λ̃ to 0-curves and let R̄ be the image of R̃ under π (we regard R̃ and R̄ as reduced effective divisors—note that they have strong normal crossings). Since R has the shape of a ring, so does R̃ by (i) and (ii); thus R̄ has the shape of a ring as well, and its dual graph is:

[figure: G_{1}; G_{2}; R̄_{0}; Σ_{m}]

where G_{1}, G_{2} are distinct members of the standard ruling Λ_{m} of 𝔽_{m} and Σ_{m} is the negative section of Λ_{m}. Since R̄_{0} · G_{1} = 1, R̄_{0} is a section of Λ_{m}, disjoint from Σ_{m}. It follows that −R̄ is a canonical divisor of 𝔽_{m}. Since R̃ is obtained from R (resp. R̄) by subdivisional blowing-up, the assertion follows. □

Proposition 3.2. (1) The basic affine rulings of type I of ℙ are precisely the stan-

dard affine rulings.

(2) Suppose that X satisfies (‡) and that the discriminants a_{0}, a_{1}, a_{2} of its singularities are pairwise relatively prime. If X admits a basic affine ruling of type I, then X ≅ ℙ(a_{0}, a_{1}, a_{2}).

Proof. Let Λ be a basic affine ruling of type I (of X), let G ∈ Λ_{∗} and let τ = (z, \binom{x}{c_{1}}, \binom{y}{c_{2}}) be the discrete part of (X, Λ, G). The connected components of the weighted graph

(G_{(−z)} ⊖ \binom{x}{c_{1}}) ⊖ \binom{y}{c_{2}}

have determinants c_{1}, c_{2} and zc_{1}c_{2} − c_{1}y − c_{2}x; so these must be equal to a_{j}, a_{k} and a_{i} respectively, for some permutation i, j, k of 0, 1, 2. Then τ = (z, \binom{x}{a_{j}}, \binom{y}{a_{k}}), where (x, y, z) is the unique integral solution of a_{i} = a_{j}a_{k}z − a_{j}y − a_{k}x with 0 ≤ x < a_{j} and 0 ≤ y < a_{k}. By Lemma 3.1, τ is also the discrete part of (ℙ(a_{0}, a_{1}, a_{2}), Λ_{i}, F), where Λ_{i} is one of the standard rulings of ℙ(a_{0}, a_{1}, a_{2}) and F is some element of (Λ_{i})_{∗}. Thus [X, Λ, G] and [ℙ(a_{0}, a_{1}, a_{2}), Λ_{i}, F] have the same image τ under the bijection 𝕊_{0}(‡) → 𝕋_{0}(‡) of 2.24. This proves both assertions of the proposition. □

Remark. Let Λ be an affine ruling of ℙ. Then the morphisms 𝔽_{m} ← ℙ̃ → ℙ̂ → ℙ defined in 2.22 induce a rational map ℙ → 𝔽_{m}. Let us make this rational map explicit in the case where Λ = Λ_{0} (notation as in 3.1). Recall that the discrete part of (ℙ, Λ_{0}, F) is (x_{0}, \binom{x_{1}}{a_{1}}, \binom{x_{2}}{a_{2}}) where (x_{0}, x_{1}, x_{2}) is the unique integral solution of a_{0} = a_{1}a_{2}x_{0} − a_{2}x_{1} − a_{1}x_{2} with 0 ≤ x_{1} < a_{1} and 0 ≤ x_{2} < a_{2} (in particular m = x_{0}). Let the notations Σ_{m}, G_{1}, G_{2}, R̄_{0} have the same meaning as before in this section. The divisors mG_{1} + Σ_{m}, mG_{2}+Σ_{m} and R̄_{0} are members of the linear system |mF +Σ_{m}| on 𝔽_{m}. It is not difficult to see that the transform of |mF + Σ_{m}| on ℙ is the linear system O(ma_{1}a_{2}) of curves of degree ma_{1}a_{2}. Now U = X_{0}X_{1}^{x_{2}}X_{2}^{x_{1}}, V_{1} = X_{1}^{ma_{2}} and V_{2} = X_{2}^{ma_{1}} define curves in O(ma_{1}a_{2}). Also, u_{1} = U/V_{1} and v = X_{2}^{a_{1}}/X_{1}^{a_{2}} are rational functions on ℙ that give equations respectively for R̄_{0} and G_{1} (at their intersection point) in 𝔽_{m}\(Σ_{m}∪G_{2}) ≅ 𝔸^{2}.

4. Some results on weighted pairs

Lemma 4.1. Consider a linear weighted pair L = (0, −1, ω_{1}, …, ω_{n}), where n ≥ 1 and ω_{j} ≤ −2 for all j and where the distinguished vertex is the one of weight 0. Let i ∈ {1, …, n} and let x, y ∈ ℤ be such that x + y = ω_{i} and x ≤ −2. Then there exists a unique column \binom{p}{c} ∈ T such that the weighted pair L\binom{p}{c} contracts to:

(3)  (ω_{1}, …, ω_{i−1}, x, 0, y, ω_{i+1} …, ω_{n}),

where the distinguished vertex is the one of weight 0, L [figure] \binom{p}{c} = (ω_{1}, …, ω_{i−1}, x) and L [figure] \binom{p}{c} contracts to (y, ω_{i+1}, …, ω_{n}). Remark. We will refer to \binom{p}{c} as “the column determined by L, i, x and y, as in 4.1”. Notation 4.2. The following conventions are used in the proof of Lemma 4.1.

(1) Write C = (c_{1}, …, c_{m}) to indicate that C is the linear chain [figure: …; c_{1}; c_{m}] (c_{i} ∈ ℤ). To indicate that we have a string of n consecutive −2, say c_{i+1} = ⋯ = c_{i+n} = −2, we may write C = (c_{1}, …, c_{i}, [n], c_{i+n+1}, …). Note that each admissible chain has a unique representation of the form ([n_{0}], z_{1}, [n_{1}], …, z_{h}, [n_{h}]), with h ≥ 0, n_{i} ≥ 0 and z_{i} ≤ −3.

(2) Consider a blowing-up C ← C′ of weighted pairs, where the underlying weighted graph of C is (c_{1}, …, c_{m}). The notation C = (c_{1}, …, c_{i−1}, c_{i}^{∗}, c_{i+1}, …, c_{m}), where ∗ is one of the three symbols ℓ, r, s, means:

(a) The distinguished vertex of C is the one of weight c_{i}.

(b) If ∗ = ℓ (resp. ∗ = r, ∗ = s) then C is blown-up at the edge [figure: c_{i−1}; c_{i}] (resp.

the edge [figure: c_{i}; c_{i+1}], the vertex ^{^{c_{i}}}). Note that ℓ, r and s remind us of “left”, “right” and “sprouting” respectively.

(When ∗ is not one of ℓ, r, s, but is really just “∗”, we mean only (a).)

(3) Suppose that we blow-up a weighted pair G_{0} according to some tableau, thus producing a sequence G_{0} ← ⋯ ← G_{N} of blowings-up. Suppose that for some k < N the graph

G_{k} = (…, c_{i−1}, c_{i}^{∗}, c_{i+1}, …) (∗ ∈ {ℓ, r, s})

has a weight c_{j} = −1 (where j ≠ i), and let \overline{G}_{k} be the contraction of G_{k} at the vertex of weight c_{j}. If one of the following holds:

(a) |j − i| > 1;

(b) j = i + 1 and ∗ ≠ r;

(c) j = i − 1 and ∗ ≠ ℓ, we say that the contraction G_{k} ≥ \overline{G}_{k} is “allowed”. In that case, the blowings-up G_{k} ← ⋯ ← G_{N} can be performed on \overline{G}_{k}, giving \overline{G}_{k} ← ⋯ ← \overline{G}_{N}, and we have a contraction of weighted pairs G_{N} ≥ \overline{G}_{N}.

Proof of Lemma 4.1. We use the conventions of 4.2. If \binom{p}{c} exists then

L\binom{p}{c} = (ω_{1}, …, ω_{i−1}, x, −1^{∗}, …),

so 2.12 implies that det(ω_{1}, …, ω_{i−1}, x) = c and det(ω_{1}, …, ω_{i−1}) = c − p′, where p′ ∈ \{1, …, c − 1\} is the inverse of p modulo c. Thus \binom{p}{c} is unique, if it exists.

To show that \binom{p}{c} exists, it suffices to construct a sequence L_{0} ← ⋯ ← L_{k} of blowings-up of weighted pairs satisfying (where e_{j} is the distinguished vertex of L_{j}):

(i) L_{0} = L; (ii) L_{0} ← L_{1} is the blowing-up at e_{0} and, for each j > 0, L_{j} ← L_{j+1} is a blowing-up

at an edge incident to e_{j}; (iii) L_{k} contracts to (3) in such a way that the following holds: if A and B are

the branches of L_{k} at e_{k}, where B contains the vertices of L, then A = (ω_{1}, …, ω_{i−1}, x) and B contracts to (y, ω_{i+1}, …, ω_{n}).

Consider the natural number N = N(L, i) = | {j < i | ω_{j} ≤ −3} |. If N = 0 then

(4)  L = (0^{s}, −1, [m], ω_{i}, …, ω_{n}),

where m = i − 1 ≥ 0, and this contracts to ((m + 1)^{s}, ω_{i} + 1, …, ω_{n}). Performing a blow-up of type “s” followed by m + 1 blows-up of type “r” gives:

([m], −2, −1^{ℓ}, −1, ω_{i} + 1, …, ω_{n}).

This contracts to ([m], −2, 0^{ℓ}, ω_{i} + 2, …, ω_{n}), which is the desired tree (3) if x = −2. If x < −2 then performing −2 − x > 0 blows-up of type “ℓ” gives:

([m], x, −1^{∗}, [−3 − x], −1, ω_{i} + 2, ω_{i+1}, …, ω_{n}),

which contracts to

([m], x, 0^{∗}, ω_{i} − x, ω_{i+1}, …, ω_{n}).

This proves the case N = 0.

If N > 0 then we may write L = (0^{s}, −1, ω_{1}, …, ω_{j}, [m], ω_{i}, …, ω_{n}), where ω_{j} ≤ −3 and m = i−j −1 ≥ 0. Since N(L, j) = N −1, there exists (by induction, with y = −1) a column \binom{p_{1}}{c_{1}} such that L\binom{p_{1}}{c_{1}} contracts to

(4’)  (ω_{1}, …, ω_{j−1}, ω_{j} + 1, 0^{ℓ}, −1, [m], ω_{i}, …, ω_{n}).

Note how (4’) is similar to (4) and let us apply the above argument to (4’). We may contract (4’) to

(ω_{1}, …, ω_{j−1}, ω_{j} + 1, (m + 1)^{ℓ}, ω_{i} + 1, …, ω_{n})

and perform a blow-up of type “ℓ” followed by m + 1 blows-up of type “r”:

(ω_{1}, …, ω_{j}, [m], −2, −1^{ℓ}, −1, ω_{i} + 1, …, ω_{n}).

This contracts to (ω_{1}, …, ω_{j}, [m], −2, 0^{ℓ}, ω_{i} + 2, …, ω_{n}), which is the desired tree if x = −2. If x < −2, perform −2 − x > 0 blows-up of type “ℓ”. □

Definition 4.3. Consider a linear weighted pair L = (0, −1, ω_{1}, …, ω_{n}), where n ≥ 0, ω_{j} ≤ −2 for all j and where the distinguished vertex is the one of weight 0. We define tableaux cont(L, ν; x, y) ∈ T for certain values of ν, x, y ∈ ℤ. The first case is:

cont(L, 0; x, −1) = 1 (the empty tableau) for all x ∈ ℤ.

Write {i | 1 ≤ i ≤ n and ω_{i} ≤ −3} = {i_{1}, …, i_{h}} (1 ≤ i_{1} < ⋯ < i_{h} ≤ n). Given (ν; x, y) ∈ ℤ^{3} satisfying

(5)  1 ≤ ν ≤ h, x ≤ −2, y ≤ −1 and x + y = ω_{i_{ν}},

let \binom{p}{c} be the unique column determined by L, i = i_{ν}, x and y as in Lemma 4.1. Then define

cont(L, ν; x, y) = \binom{p}{c}.

We also define a subset Cont(L) of T by

Cont(L) = {1} ∪ {cont(L, ν; x, y) | (ν; x, y) satisfies (5)}

and a map Cont(L) → Cont(L^{t}) (C ↦ C̃) by:

C̃ = \{ \begin{matrix}cont(L^{t}, h − ν; x′, −1) (for suitable x′), & if C = cont(L, ν; x, −1); \\ cont(L^{t}, h − ν + 1; y, x), & if C = cont(L, ν; x, y) and y ≤ −2. \end{matrix}

This makes sense because, given L and C ∈ Cont(L) \ {1}, there is a unique triple (ν; x, y) satisfying cont(L, ν; x, y) = C. Note that 1 ↦ cont(L^{t}, h; x′, −1) (for suitable x′) and cont(L, h; x, −1) ↦ 1.

We call C̃ the L-dual of C. It is easily verified that Cont(L) → Cont(L^{t}) is bijective and that its inverse is C ↦ L^{t}-dual of C. Lemma 4.4. Consider a linear weighted pair L = (0, −m, ω_{1}, …, ω_{n}), where m ∈ ℤ, n ≥ 0, ω_{j} ≤ −2 and where the distinguished vertex is the one of weight 0. Let T = ( \begin{matrix}p & ··· \\ c & ··· \end{matrix}) be a tableau with at least two columns and such that \binom{p}{c} ≠ \binom{1}{1}. Suppose that the weighted graph Γ = L [figure] T contracts to an admissible chain A satisfying |A| ≤ |L|. Then m = 1 and one of the following holds:

(1) \binom{p}{c} ∈ T_{k}(L), for some k > 0;

(2) \binom{p}{c} ∈ Cont(L).

Proof. Consider the sequence of blowing-ups of linear chains

L = G_{0} ← G_{1} ← ⋯ ← G_{N} = L\binom{p}{c}

produced by blowing-up L according to \binom{p}{c}. Note that |Γ| ≥ |G_{N}| > |L| ≥ |A|, so Γ contains a vertex of weight −1. Since \binom{p}{c} ≠ \binom{1}{1}, this implies that m = 1. Using the conventions of 4.2, we may write

(6)  L = G_{0} = (0^{s}, −1, [n_{0}], z_{1}, [n_{1}], …, z_{h}, [n_{h}]),

where h ≥ 0, z_{j} ≤ −3 and n_{j} ≥ 0; also, it is allowed (4.2.3) to contract (6) to:

(7)  ((n_{0} + 1)^{s}, z_{1} + 1, [n_{1}], …).

Since Γ contracts to an admissible chain, the number n_{0} + 1 must be decreased by blowing-up until it becomes negative, i.e., the next n_{0} + 2 trees must be:

(8)  (−1^{r}, n_{0}, z_{1} + 1, [n_{1}], …), …, ([n_{0}], −2, −1^{∗}, −1, z_{1} + 1, [n_{1}], …),

where ∗ ∈ {ℓ, r, s}. Since the last chain in (8) contains |L| + 1 vertices, the condition |A| ≤ |L| implies that ∗ ≠ r. Then we may contract the last chain to

(9)  ([n_{0}], −2, 0^{∗}, z_{1} + 2, [n_{1}], …) (with ∗ ∈ {ℓ, s}).

In the special case where h = 0, the chains (7) and (9) are simply ((n_{0} + 1)^{s}) and ([n_{0}], −2, 0^{∗}) (with ∗ ∈ \{ℓ, s\}) respectively, and the latter implies that L\binom{p}{c} shrinks to ([n_{0}], x, 0^{∗}) for some x ≤ −2; let k = −1 − x > 0 then L\binom{p}{c}\binom{1}{1}^{k} ≈ L^{t}, so condition (1) holds. So we may assume that h > 0. Then, by (9), there exists j ≤ N such that G_{j} contracts to a chain of the form:

(10)  ([n_{0}], …, [n_{i−1}], x, 0^{∗}, y, [n_{i}], …, [n_{h}]) (with ∗ ∈ {ℓ, s})

where 1 ≤ i ≤ h, x ≤ −2 and x + y = z_{i}. Let us assume that j is maximal with respect to this property. Note that y < 0, because ∗ ≠ r and Γ shrinks to an admissible chain. It suffices to prove:

Claim. j = N or L\binom{p}{c} ≥ ([n_{0}], z_{1}, [n_{1}], …, z_{h}, [n_{h}], x, 0^{∗}) for some x ≤ −2.

Indeed, if j = N then \binom{p}{c} = cont(L, i; x, y) and if L\binom{p}{c} ≥ ([n_{0}], z_{1}, [n_{1}], …, z_{h}, [n_{h}], x, 0^{∗}) then L\binom{p}{c}\binom{1}{1}^{k} ≈ L^{t}, with k = −1 − x > 0.

To prove the claim, we may assume that j < N; then ∗ ≠ s in (10), so ∗ = ℓ and the tree which immediately follows (10) is:

(11)  ([n_{0}], …, [n_{i−1}], x − 1, −1^{⋆}, −1, y, [n_{i}], …, [n_{h}]) (with ⋆ ∈ {ℓ, r, s}).

Note that ⋆ = r, otherwise it would be allowed to shrink (11) to

([n_{0}], …, [n_{i−1}], x − 1, 0^{⋆}, y + 1, [n_{i}], …, [n_{h}]) (with ⋆ ∈ {ℓ, s}),

contradicting the assumption that j is maximal. Since the chain (11) contains |L| + 1 vertices and ⋆ = r, it follows that y = −1 (and x = z_{i} + 1), so (10) is:

(12)  ([n_{0}], …, [n_{i−1}], z_{i} + 1, 0^{ℓ}, −1, [n_{i}], …, [n_{h}]).

Note the similarity between (12) and (6); the above argument applied to (12) shows that one of the following two conditions holds:

(i) i = h, in which case (12) contracts to ([n_{1}], …, [n_{h−1}], z_{h} + 1, (n_{h} + 1)^{ℓ}); as in the case h = 0, this implies that condition (1) is satisfied and we are done in this case.

(ii) Some G_{j′} (with j′ > j) contracts to

([n_{0}], …, [n_{i}], x′, 0^{∗}, y′, [n_{i+1}], …, [n_{h}]) (with ∗ ∈ {ℓ, s})

where i + 1 ≤ h, x′ ≤ −2 and x′ + y′ = z_{i+1}. By maximality of j, this is impossible.

This proves the claim and hence the lemma. □

5. Basic affine rulings of type II

Proposition 5.1. Let a_{0}, a_{1}, a_{2} be pairwise relatively prime positive integers and let τ ∈ 𝕋_{II.1}(a_{0}, a_{1}, a_{2}). Then τ is not minimal in 𝕋(‡) and its immediate predecessor belongs to 𝕋_{I}(a_{0}, a_{1}, a_{2}).

We have to establish two lemmas before proving this, but let us first give: Corollary 5.2. Let X be a surface of type [a, b, c], where a, b, c are pairwise relatively prime positive integers. Then every basic affine ruling of type II of X reduces to one of type I. In particular, if X admits a basic affine ruling of type II then X ≅ ℙ(a, b, c).

Proof. The last assertion follows from Proposition 3.2. Let Λ be a basic affine ruling of type II of X. Then, for some F ∈ Λ_{∗}, τ = disc(X, Λ, F) belongs to 𝕋_{II.1}(a_{0}, a_{1}, a_{2}) for some permutation a_{0}, a_{1}, a_{2} of a, b, c (see 2.27). By Proposition 5.1, there exists τ′ ∈ 𝕋_{I}(a_{0}, a_{1}, a_{2}) such that τ > τ′. We have τ′ ∈ 𝕋(X) by 2.25, so there exists an affine ruling Λ′ of X and an element F′ of Λ′_{∗} such that τ′ = disc(X, Λ′, F′); note that Λ′ is of type I. (In the language of [2], Λ′ is obtained from Λ by “reduction”.) □

Lemma 5.3. Let

A′ : [figure: …; …; e_{1}; e_{m}; f_{1}; f_{n}; x; α; y; u′]

and

A′′ : [figure: …; …; α; β; e_{1}; e_{m}; f_{1}; f_{n}; x+y; u′′]

be two linear chains, where m, n ≥ 0. If A′ and A′′ are equivalent to the same admissible chain A, then one of the linear chains X = (e_{1}, …, e_{m}, x) and Y = (y, f_{1}, …, f_{n}) shrinks to the empty graph.

Proof. If some e_{i} or f_{i} is −1, then we may blow-down A′ and A′′ at the corresponding vertex; this produces linear chains Ā′ and Ā′′ which still satisfy the hypothesis of the lemma (with possibly different values of m, n, x, y, β) and where the new X and Y are obtained from the old ones by blowing-down. We may therefore assume that e_{i} < −1 and f_{i} < −1 for all i. Since A′ contracts to an admissible chain, x, y < 0 and consequently x + y ≤ −2; since A′′ contracts to an admissible chain, α and β are negative and at most one of them is −1. Thus at most one weight in A′′ is −1.

Given a linear chain C, let w(C) denote the sum of the weights in C. Note that if we blow-down C at a vertex v of weight −1 then w(C) increases by n(v) + 1 where n(v) ∈ {0, 1, 2} is the number of neighbors of v in C.

Since A′ and A′′ have the same number of vertices and contract to the same chain A, there exist two sequences of linear chains:

S′ : A′ = A′_{0}, …, A′_{s} = A and S′′ : A′′ = A′′_{0}, …, A′′_{s} = A

(of the same length s) where each A′_{i} (resp. A′′_{i}) is obtained from A′_{i−1} (resp. A′′_{i−1}) by blowing-down one vertex v′_{i−1} (resp. v′′_{i−1}). Note that S′′ is unique and {n(v′′_{i−1})}_{i=1}^{s} is nonincreasing; also, we may choose S′ in such a way that {n(v′_{i−1})}_{i=1}^{s} is nonincreasing.

Note that β < 0 implies that w(A′′) < w(A′), so

w(A′_{s}) − w(A′_{0}) < w(A′′_{s}) − w(A′′_{0}).

So there exists j ∈ {1, …, s} such that n(v′_{j−1}) = 1 and n(v′′_{j−1}) = 2. In particular n(v′′_{0}) = 2, so β = −1 and α < −1. Note that the vertex u′′ is still present in A′′_{j} and that its weight there is α + j, which implies that α + j < 0. Consequently, u′ is still present in A′_{j}; since n(v′_{j−1}) = 1, this implies that one of X, Y contracts to the empty graph. □

Lemma 5.4. Let L = (0, −m, ω_{1}, …, ω_{n}) be a linear weighted pair such that m ≥ 1, n ≥ 0, ω_{i} ≤ −2 and the distinguished vertex is the one of weight 0. Consider a tableau T = ( \begin{matrix}p & 1 \\ c & a \end{matrix}) where \binom{p}{c} ≠ \binom{1}{1} and a ≥ 1. Suppose that the weighted graph Γ = L [figure] T is equivalent to an admissible chain and that, for some β < 0, Γ is also equivalent to one of:

C = (−a, β, ω_{1}, …, ω_{n}), C′ = (−a, β, ω_{n}, …, ω_{1}).

Then m = 1 and \binom{p}{c} ∈ T_{k}(L) for some k ≥ 0.

Proof. Consider the admissible chain A which is equivalent to Γ. Since A is equivalent to C or C′, we have |A| ≤ |L|; so Lemma 4.4 implies that m = 1 and that \binom{p}{c} belongs to either T_{k}(L) (some k > 0) or Cont(L). So we may assume that \binom{p}{c} ∈ Cont(L); then L\binom{p}{c} contracts to

(ω_{1}, …, ω_{i−1}, x, 0, y, ω_{i+1}, …, ω_{n})

and Γ contracts to

(ω_{1}, …, ω_{i−1}, x, −a, y, ω_{i+1}, …, ω_{n}),

where 1 ≤ i ≤ n, x ≤ −2 and x+y = ω_{i}. Using Lemma 5.3 and again the fact that A is equivalent to C or C′, we conclude that one of (ω_{1}, …, ω_{i−1}, x), (y, ω_{i+1}, …, ω_{n}) shrinks to the empty graph. Since x and all ω_{j} are strictly less than −1, (y, ω_{i+1}, …, ω_{n}) shrinks to the empty graph. Since L\binom{p}{c} contracts to (ω_{1}, …, ω_{i−1}, x, 0, y, ω_{i+1}, …, ω_{n}), where the distinguished vertex is the one of weight 0, we conclude that L\binom{p}{c} contracts to a linear weighted pair, so \binom{p}{c} ∈ T_{k}(L) for some k. □

Proof of Proposition 5.1. Write τ = (m, T_{1}, T_{2}) with T_{1} = ( \begin{matrix}p \\ a_{1} \end{matrix} ) and T_{2} = ( \begin{matrix}p_{2} & 1 \\ c_{2} & a_{2} \end{matrix} ) (see 2.28). For i = 0, 1, 2, define A_{i} and G_{i} as in 2.26; then det(G_{0}) = a_{0} because τ ∈ 𝕋_{II.1}(a_{0}, a_{1}, a_{2}); also, a calculation using 2.12 and 1.11 gives det(G_{0}) = a_{2}c_{2}Δ − a_{1}, where we define Δ = mc_{2}a_{1} − c_{2}p − a_{1}p_{2}. In particular, a_{2} divides a_{0} + a_{1}. Let us record

(13)  T_{1} = ( \begin{matrix}p \\ b \end{matrix}) , T_{2} = ( \begin{matrix}p_{2} & 1 \\ c_{2} & a \end{matrix}) and a | b + c,

where we define a, b, c by (c, b, a) = (a_{0}, a_{1}, a_{2}). Note that m ≥ 1, c_{2} > p_{2} ≥ 1, a ≥ 1 and b > p ≥ 0 are integers and gcd(p_{2}, c_{2}) = 1 = gcd(p, b). Also, \binom{p}{b} is subject to 2.29. We may write G_{0} = L [figure] T_{2}, where L is the weighted pair G_{(−m)} [figure] \binom{p}{b}:

(14)  [figure: L :; …; 0; −m; ω_{1}; ω_{n}; _{p}; _{b}]

where the leftmost vertex is the distinguished one, and where we used 2.12 for computing the determinants.

Claim. There exists a linear chain C^{∗} and an integer γ > 0 satisfying

(15)  C^{∗} contracts to A_{0}

and

(16)  C^{∗} = (−a, −γ, ω_{1}, …, ω_{n}) or C^{∗} = (−a, −γ, ω_{n}, …, ω_{1}).

The proof of the Claim splits into two cases.

Case b > 1. By 2.12, the subdiscriminants of G_{1} = Z [figure] \binom{p}{b} are b−p and b−p′, where p′ is defined by \binom{p′′}{p′} = \binom{p}{b}^{∗} (Z denotes the weighted pair consisting of a single vertex of weight zero). On the other hand, τ ∈ 𝕋_{II.1}(c, b, a) implies that G_{1} has subdiscriminants b′(a, c) and b′(c, a), so {b − p, b − p′} = {b′(a, c), b′(c, a)}; for later use, we record:

(17)  b − b′ ∈ {p, p′}, where b′ = b′(a, c).

Observe that b + c − ab′ ≡ c − ab′ ≡ 0 (mod b) by definition of b′, and b + c − ab′ ≡ b + c ≡ 0 (mod a) by (13); since a, b are relatively prime, b + c − ab′ = (γ − 1)ab (some γ ∈ ℤ), so c = (γ − 1)ab + ab′ − b. Since c ≥ 1, we have γ ≥ 1. Let us define

c̄ = γb − (b − b′)

then c̄ > 0 and we have equations (i) and (ii) in:

(i)  c = ac̄ − b
(ii) c̄ = γb − (b − b′)
(iii)  b = q_{1}(b − b′) − r_{2}
b − b′ = q_{2}r_{2} − r_{3}
⋮
(iv)  r_{s−1} = q_{s}r_{s} − r_{s+1}

where equations (iii)–(iv) are the outer euclidean algorithm on r_{0} = b and r_{1} = b − b′ (r_{i}, q_{i} ∈ ℕ, r_{i−1} = q_{i}r_{i} − r_{i+1}, 0 ≤ r_{i+1} < r_{i}, r_{s+1} = 0). The integers q_{i} are now used to define a linear chain

(18)  [figure: C^{∗} :; …; −a; −γ; −q_{1}; −q_{s}; _{b−b′}; _{b}; _{c̄}; _{c}]

with determinants as indicated. Note that, in C^{∗}, all weights are negative and at most one is −1 (q_{i} ≥ 2 for all i and if a = 1 = γ then equations (i) and (ii) give c = b′−b < 0, a contradiction); this and det(C^{∗}) > 0 imply that C^{∗} shrinks to an admissible chain. Since C^{∗} and A_{0} have the same discriminant c and, modulo c, have a subdiscriminant in common (Equation (i) gives c̄ ≡ c′(a, b) (mod c)), 1.15 implies that (15) holds. By (14), (18) and (17), we have that (−q_{1}, …, −q_{s}) is (ω_{1}, …, ω_{n}) or (ω_{n}, …, ω_{1}), so (16) holds.

Case b = 1. Define γ = (b + c)/a then, by (13), γ is a positive integer. Let C^{∗} be the linear chain (−a, −γ), then det(C^{∗}) = γa − 1 = γa − b = c > 0 and it is easy to see that C^{∗} shrinks to an admissible chain. Since C^{∗} and A_{0} have the same discriminant c and, modulo c, have a subdiscriminant in common (γ ≡ c′(a, b) (mod c)), 1.15 implies that (15) holds. We have n = 0 in (14), so (16) holds and the above Claim is proved.

Now (15), (16) and Lemma 5.4 imply that m = 1 and that \binom{p_{2}}{c_{2}} ∈ T_{k}(L) for some k ≥ 0. By 2.19, τ is non-minimal in 𝕋(‡) and we may consider its immediate predecessor τ^{−}. By 2.31, τ^{−} ∈ 𝕋_{I}(a_{0}, a_{1}, a_{2}). □

6. Basic affine rulings of type III

Lemma 6.1. Consider a linear weighted pair L = (0, −1, ω_{1}, …, ω_{n}), where n ≥ 0, ω_{j} ≤ −2 for all j and where the distinguished vertex is the one of weight 0. Consider an element C of Cont(L) and its L-dual C̃ ∈ Cont(L^{t}).

(1) LC ≈ L^{t}C̃ (equivalence of weighted pairs).

(2) L [figure] C ∼ L^{t} [figure] C̃ and L [figure] C ∼ L^{t} [figure] C̃ (equivalences of weighted graphs).

(3) Write C = \binom{p}{c} and C̃ = (p̃ ; c̃), using the convention of 2.29 if necessary. Then c = det(L [figure] C) and c̃ = det(L [figure] C).

Proof. We prove assertions (1) and (2) simultaneously. Let i_{1} < ⋯ < i_{h} be as in 4.3 and write z_{j} = ω_{i_{j}} for j = 1, …, h. Then

L = (0, −1, [n_{0}], z_{1}, [n_{1}], …, z_{h}, [n_{h}]) and L^{t} = (0, −1, [n_{h}], z_{h}, …, z_{1}, [n_{0}])

for some integers n_{j} ≥ 0. If h = 0 then L = L^{t} and C = 1 = C̃, so (1) is trivial in this case; also, L [figure] C = (−1, [n]) ∼ L^{t} [figure] C̃, since L^{t} [figure] C̃ is the empty graph; similarly, L^{t} [figure] C̃ ∼ L [figure] C, so (1) and (2) hold in this case. Assume h > 0.

If C = cont(L, ν; x, −1) then

LC ≥ ([n_{0}], …, [n_{ν−1}], z_{ν} + 1, 0^{∗}, −1, [n_{ν}], z_{ν+1}, [n_{ν+1}], …, [n_{h}])

≥ ([n_{0}], …, [n_{ν−1}], z_{ν} + 1, (n_{ν} + 1)^{∗}, z_{ν+1} + 1, [n_{ν+1}], …, [n_{h}]); (19)

since C̃ = cont(L^{t}, h − ν; x′, −1), we also have:

(20)  L^{t}C̃ ≥ ([n_{h}], …, [n_{ν+1}], z_{ν+1} + 1, 0^{∗}, −1, [n_{ν}], z_{ν}, [n_{ν−1}], …, [n_{0}])

≥ ([n_{h}], …, [n_{ν+1}], z_{ν+1} + 1, (n_{ν} + 1)^{∗}, z_{ν} + 1, [n_{ν−1}], …, [n_{0}]). (21)

Since the weighted pairs (19) and (21) are the same, LC ≈ L^{t}C̃. This also shows that

L [figure] C ≥ (−1, [n_{ν}], z_{ν+1}, [n_{ν+1}], …, [n_{h}]) ≥ (z_{ν+1} + 1, [n_{ν+1}], …, [n_{h}]) = L^{t} [figure] C̃

and

L^{t} [figure] C̃ ≥ (−1, [n_{ν}], z_{ν}, [n_{ν−1}], …, [n_{0}]) ≥ (z_{ν} + 1, [n_{ν−1}], …, [n_{0}]) = L [figure] C,

so (2) holds as well.

If C = cont(L, ν; x, y) with y ≤ −2, then

LC ≥ ([n_{0}], …, [n_{ν−1}], x, 0^{∗}, y, [n_{ν}], z_{ν+1}, [n_{ν+1}], …, [n_{h}]);

since C̃ = cont(L^{t}, h − ν + 1; y, x), we also have:

L^{t}C̃ ≥ ([n_{h}], …, [n_{ν}], y, 0^{∗}, x, [n_{ν−1}], z_{ν−1}, …, [n_{0}]).

So we have LC ≈ L^{t}C̃,

L [figure] C ≥ (y, [n_{ν}], z_{ν+1}, [n_{ν+1}], …, [n_{h}]) = L^{t} [figure] C̃

and

L^{t} [figure] C̃ ≥ (x, [n_{ν−1}], z_{ν−1}, …, [n_{0}]) = L [figure] C,

so (1) and (2) hold in all cases.

We already know that c = det(L [figure] C): this follows from 2.12 and was observed at the beginning of the proof of Lemma 4.1 (det(ω_{1}, …, ω_{i−1}, x) = c). Applying this fact to L^{t} gives c̃ = det(L^{t} [figure] C̃), and this is equal to det(L [figure] C) by part (2). □

Lemma 6.2. Let τ = (m, T_{1}, T_{2}) be a minimal element of 𝕋(‡), where T_{i} = ( \begin{matrix}p_{i} & 1 \\ c_{i} & a_{i} \end{matrix}) ∈ T (i = 1, 2). Suppose that the weighted graph (G_{(−m)} [figure] T_{1}) [figure] T_{2} shrinks to a graph with at most |G_{(−m)} [figure] T_{1}| vertices. Then m = 1 and if we write L = G_{(−1)} [figure] T_{1} then:

(1) \binom{p_{2}}{c_{2}} ∈ Cont(L) and its L-dual is not the empty tableau.

From now-on, let (p̃_{2} ; c̃_{2}) ∈ Cont(L^{t}) denote the L-dual of \binom{p_{2}}{c_{2}}, define T̃_{2} = ( \begin{matrix}p̃_{2} & 1 \\ c̃_{2} & a_{2} \end{matrix} ) ∈ T and τ̃ = (1, Ť_{1}, T̃_{2}). Then:

(2) τ ≡ τ̃ and τ̃ is a minimal element of 𝕋(‡).

(3) (G_{(−1)} [figure] T_{1}) [figure] T_{2} ∼ (G_{(−1)} [figure] Ť_{1}) [figure] T̃_{2}.

(4) c_{2} + c̃_{2} = a_{1}c_{1}Δ(τ), where Δ(τ) = mc_{1}c_{2} − c_{1}p_{2} − c_{2}p_{1} = c_{1}c_{2} − c_{1}p_{2} − c_{2}p_{1}.

(5) p̃_{2} = −c_{2} + p_{2} + a_{1}p_{1}Δ(τ).

(6) Δ(τ) = Δ(τ̃).

(7) If τ ∈ 𝕋_{III}(a_{0}, a_{1}, a_{2}), for some pairwise relatively prime positive integers a_{0}, a_{1}, a_{2}, then τ̃ ∈ 𝕋_{III}(a_{0}, a_{1}, a_{2}).

Proof. Since τ ∈ 𝕋(‡), the intersection matrix of Γ = L [figure] T_{2} is negative definite; thus Γ contracts to an admissible chain A, and |A| ≤ |L| by the assumption. By Lemma 4.4, m = 1 and \binom{p_{2}}{c_{2}} belongs to either T_{k}(L) (for some k > 0) or Cont(L). By 2.19 and minimality of τ, we have in fact \binom{p_{2}}{c_{2}} ∉ T_{k}(L) (for all k ∈ ℕ), so \binom{p_{2}}{c_{2}} ∈ Cont(L). If the L-dual of \binom{p_{2}}{c_{2}} is empty then L\binom{p_{2}}{c_{2}} ≈ L^{t} by Lemma 6.1, so L\binom{p_{2}}{c_{2}} contracts to a linear weighted pair, so \binom{p_{2}}{c_{2}} ∈ T_{k}(L) for some k ∈ ℕ (by 2.15) and this contradicts an earlier observation. So assertion (1) holds.

If τ̃ is non-minimal then (2.19) (p̃_{2} ; c̃_{2}) ∈ T_{k}(L^{t}) for some k, so (2.15) L^{t}(p̃_{2} ; c̃_{2}) contracts to a linear weighted pair, so (6.1) L\binom{p_{2}}{c_{2}} has the same property, so (2.15) \binom{p_{2}}{c_{2}} ∈ T_{k}(L) for some k, a contradiction. Hence, τ̃ is minimal. Lemma 6.1 implies

(22)  (G_{(−1)} [figure] T_{1})\binom{p_{2}}{c_{2}} = L\binom{p_{2}}{c_{2}} ≈ L^{t}(p̃_{2} ; c̃_{2}) = (G_{(−1)} [figure] Ť_{1})(p̃_{2} ; c̃_{2})

and (G_{(−1)} ⊖ T_{1})\binom{p_{2}}{c_{2}} ≈ (G_{(−1)} ⊖ Ť_{1})(p̃_{2} ; c̃_{2}) easily follows; “multiplying” both sides by \binom{1}{a_{2}} gives (G_{(−1)} ⊖ T_{1})T_{2} ≈ (G_{(−1)} ⊖ Ť_{1})T̃_{2}, i.e., assertion (2) holds.

If P ≈ P′ are equivalent weighted pairs and T is a tableau, then P [figure] T ∼ P′ [figure] T. Applying this to (22) (with T = \binom{1}{a_{2}}) gives assertion (3).

To prove assertion (4), note that L = G_{(−1)} [figure] T_{1} is as follows:

[figure: L :; …; 0; u; −1; v; ω_{1}; ω_{n}; _{a_{1}c_{1}p_{1}+1}; _{a_{1}c_{1}^{2}}]

and Lemma 2.12 gives:

L\binom{p_{2}}{c_{2}} : … ^{}^{}… ^{}^{}^{}^{}… ^{}^{−1}  ^{u}  ^{−1}^{v}  ^{ω_{1}}  ^{ω_{2}}  ^{ω_{n}} _{a_{1}c_{1}p_{1}+1} _{a_{1}c_{1}^{2}} _{p_{2}} _{c_{2}} \overbrace{L [figure] \binom{p_{2}}{c_{2}}}

We have c̃_{2} = det(L [figure] \binom{p_{2}}{c_{2}}) by Lemma 6.1, so 1.11 gives

c̃_{2} = c_{2}a_{1}c_{1}^{2} − c_{2}(a_{1}c_{1}p_{1} + 1) − p_{2}a_{1}c_{1}^{2} = −c_{2} + a_{1}c_{1}(c_{1}c_{2} − c_{1}p_{2} − c_{2}p_{1})

and assertion (4) holds.

Observe that τ̃ satisfies the hypothesis of the Lemma and that assertion (4) gives c̃_{2} + ˜ c̃_{2} = a_{1}c_{1}Δ(τ̃); since ˜ c̃_{2} = c_{2}, we obtain a_{1}c_{1}Δ(τ) = a_{1}c_{1}Δ(τ̃), so assertion (6) holds. Then (6) gives:

c_{1}c_{2} − c_{1}p_{2} − c_{2}p_{1} = c_{1}c̃_{2} − c_{1}p̃_{2} − c̃_{2}(c_{1} − p_{1})
= p_{1}c̃_{2} − c_{1}p̃_{2}
= p_{1}(−c_{2} + a_{1}c_{1}Δ(τ)) − c_{1}p̃_{2},

so c_{1}c_{2} −c_{1}p_{2} = p_{1}a_{1}c_{1}Δ(τ)−c_{1}p̃_{2} and (5) follows from this. In view of 2.30, (7) follows from (2) and (3). □

The set E

We will now define a subset E of 𝕋(‡) and show that its elements can be constructed from those which are not minimal in (𝕋(‡), <). Definition 6.3. Let E be the set of triples τ = (m, T_{1}, T_{2}) ∈ 𝕋(‡) satisfying:

(1) For each i = 1, 2, T_{i} satisfies condition 2.16.3: T_{i} = ( \begin{matrix}p_{i} & 1 \\ c_{i} & a_{i} \end{matrix});

(2) the weighted graph (G_{(−m)} [figure] T_{1}) [figure] T_{2} shrinks to an admissible chain containing at most four vertices;

(3) Δ(τ) ≠ 1 or min(a_{1}, a_{2}) ≠ 1, where Δ(τ) = mc_{1}c_{2} − c_{1}p_{2} − p_{1}c_{2}. 6.4. Let τ = (m, T_{1}, T_{2}) ∈ E.

(1) m = 1, because (G_{(−m)} [figure] T_{1}) [figure] T_{2} contains at least 7 vertices and hence must contain a vertex of weight −1.

(2) If τ is minimal in 𝕋(‡) then τ satisfies the hypothesis of 6.2. In particular, τ̃ is

defined and minimal, and we also have τ̃ ∈ E by parts (3) and (6) of 6.2. 6.5. Let τ = (1, T_{1}, T_{2}) ∈ E, with notation T_{i} = ( \begin{matrix}p_{i} & 1 \\ c_{i} & a_{i} \end{matrix}) as before. For i = 1, 2, consider the vertex e_{i} of Γ = (G_{(−1)} [figure] T_{1}) [figure] T_{2} which is the last vertex created by the blowing-up according to \binom{p_{i}}{c_{i}}:

(23)  Γ:  _{}_{}…  …^{−a_{2}−1}_{e_{2}}  ^{−a_{1}−1}_{e_{1}} \overbrace{Δ̃} \overbrace{L [figure] \binom{p_{2}}{c_{2}}}  \overbrace{G_{(−1)} [figure] \binom{p_{1}}{c_{1}}} \overbrace{L [figure] \binom{p_{2}}{c_{2}}}

where L = G_{(−1)} [figure] T_{1}, Δ̃ = (G_{(−1)} [figure] \binom{p_{1}}{c_{1}}) [figure] \binom{p_{2}}{c_{2}} and det(Δ̃) = Δ(τ).

We claim that at least one of e_{1}, e_{2} disappears in the shrinking process which transforms Γ into an admissible chain A such that |A| ≤ 4. Indeed, the subtrees B_{1} = G_{(−1)} [figure] \binom{p_{1}}{c_{1}} and B_{2} = L [figure] \binom{p_{2}}{c_{2}} are nonempty (because \binom{p_{i}}{c_{i}} ≠ \binom{1}{1}) and at least one of them contains more than one vertex (otherwise p_{i} = c_{i} − 1 for each i = 1, 2, so Δ(τ) = c_{1}c_{2} − c_{1}(c_{2} − 1) − (c_{1} − 1)c_{2} ≤ 0, which is absurd); since the shrinking is initiated in Δ̃, if no e_{i} disappears then |A| ≥ |B_{1}| + |B_{2}| + 2 ≥ 5, a contradiction. Note, also, that the shrinking process is unique, i.e., the order in which the vertices disappear is well-defined. This allows us to give:

Definition 6.6. We denote by E^{+} the set of τ ∈ E for which e_{1} disappears before e_{2} (or e_{1} disappears but e_{2} does not). Given τ = (1, T_{1}, T_{2}) ∈ E, let τ^{×} = (1, T_{2}, T_{1}). Then τ^{×} ∈ E and exactly one of τ, τ^{×} is in E^{+}. Lemma and definition 6.7. Let τ = (1, T_{1}, T_{2}) ∈ E, with notation T_{i} = ( \begin{matrix}p_{i} & 1 \\ c_{i} & a_{i} \end{matrix}).

(1) If τ is non-minimal in (𝕋(‡), <), then τ ∈ E^{+}.

(2) τ ∈ E^{+} if and only if c_{1} < c_{2}. Given τ ∈ E minimal in (𝕋(‡), <), define τ^{∗} = (τ̃)^{×}. By 6.4, τ^{∗} is defined and belongs to E.

(3) If τ ∈ E is minimal in (𝕋(‡), <) then τ ∈ E^{+} ⟺ τ^{∗} ∈ E^{+}. Moreover, if

τ ∈ E^{+} then c_{1}^{∗} < c_{2}^{∗} = c_{1} < c_{2}, where we write τ^{∗} = (1, ( \begin{matrix}p_{1}^{∗} & 1 \\ c_{1}^{∗} & a_{1}^{∗} \end{matrix} ) , ( \begin{matrix}p_{2}^{∗} & 1 \\ c_{2}^{∗} & a_{2}^{∗} \end{matrix} )).

(4) If τ ∈ E is minimal in (𝕋(‡), <), and if τ ∈ 𝕋_{III}(a_{0}, a_{1}, a_{2}) for some pairwise

relatively prime positive integers a_{0}, a_{1}, a_{2}, then τ^{∗} ∈ 𝕋_{III}(a_{0}, a_{2}, a_{1}). Given τ ∈ E^{+}, define τ^{⋆} = (τ^{×})^{∼}.

(5) If τ ∈ E^{+} then τ^{⋆} is defined, belongs to E^{+} and is minimal in (𝕋(‡), <).

(6) If τ ∈ E^{+} then (τ^{⋆})^{∗} = τ and, if τ is minimal, (τ^{∗})^{⋆} = τ.

Proof. Let L = G_{(−1)} [figure] T_{1}.

Suppose that τ is non-minimal. Then, by 2.19, \binom{p_{2}}{c_{2}} ∈ T_{k}(L) for some k ∈ ℕ, so (2.15) the weighted pair L\binom{p_{2}}{c_{2}} contracts to a linear weighted pair. Equivalently, the tree Δ̃ ∪ \{e_{1}\} ∪ (G_{(−1)} [figure] \binom{p_{1}}{c_{1}}) is equivalent to the empty graph (see the picture (23) in 6.5). In particular e_{1} disappears before e_{2}, so τ ∈ E^{+}, which proves assertion (1). Let us continue and show that c_{1} < c_{2} in this case. By 5.38 of [2] we have

M(L) = ( \begin{matrix}a_{1}p_{1}(c_{1}−p_{1})−1 & a_{1}c_{1}^{2}−a_{1}c_{1}p_{1}−1 \\ a_{1}c_{1}p_{1}−1 & a_{1}c_{1}^{2} \end{matrix} ), so \binom{p_{2}}{c_{2}} ∈ T_{k}(L) implies:

(24)  \binom{p_{2}}{c_{2}} = M(L)\binom{1}{k} = \binom{a_{1}p_{1}(c_{1} − p_{1}) − 1}{a_{1}c_{1}p_{1} − 1} + k\binom{a_{1}c_{1}^{2} − a_{1}c_{1}p_{1} − 1}{a_{1}c_{1}^{2}}.

Consequently, if c_{1} ≥ c_{2} then k = 0 and

(25)  \binom{p_{2}}{c_{2}} = \binom{a_{1}p_{1}(c_{1} − p_{1}) − 1}{a_{1}c_{1}p_{1} − 1},

so c_{1} ≥ c_{2} = a_{1}c_{1}p_{1} − 1, so (a_{1}p_{1} − 1)c_{1} ≤ 1, so a_{1} = 1 = p_{1}. Hence, T_{1} = ( \begin{matrix}1 & 1 \\ c_{1} & 1 \end{matrix} )

and T_{2} = ( \begin{matrix}c_{1}−2 & 1 \\ c_{1}−1 & a_{2} \end{matrix} ) and it follows that Δ(τ) = c_{1}(c_{1} − 1) − c_{1}(c_{1} − 2) − (c_{1} − 1)1 = 1. So, by assuming that c_{1} ≥ c_{2}, we derived that Δ(τ) = 1 = a_{1}, which contradicts the assumption that τ ∈ E. We conclude that c_{1} < c_{2} whenever τ ∈ E is non-minimal in (𝕋(‡), <).

Assume that τ ∈ E^{+} is minimal in (𝕋(‡), <). Then (6.4, 6.2) \binom{p_{2}}{c_{2}} ∈ Cont(L) and we may consider its L-dual (p̃_{2} ; c̃_{2}) ∈ Cont(L^{t}). We claim that

(26)  c̃_{2} < c_{1}.

For this argument, refer to the picture (23) in 6.5, but let the weights in G_{(−1)} [figure] \binom{p_{1}}{c_{1}} be as follows:

[figure: e_{1}; −a_{1}−1; w_{s}; …; w_{1}; …]

The shrinking of Γ = L [figure] T_{2} to an admissible chain A can be broken into two parts, Γ ≥ Γ′ ≥ A, where e_{2} is still present in Γ′ and either (i) e_{2} has weight −1 in Γ′ or (ii) Γ′ = A.

Since \binom{p_{2}}{c_{2}} ∈ Cont(L), we also have a contraction of weighted pairs

(27)  L\binom{p_{2}}{c_{2}} ≥ (…, [n_{i−1}], x, 0, y, [n_{i}], …)

(for some i, x, y) where e_{2} is the vertex of weight 0 in the right hand side. Thus the contraction (27) increases the weight of e_{2}; consequently, the weight of e_{2} is increased by the contraction Γ ≥ Γ′. It follows that all vertices of Δ̃ ∪ {e_{1}} (see (23)) disappear in the contraction Γ ≥ A, because we know that e_{1} disappears (τ ∈ E^{+}). Thus

L [figure] \binom{p_{2}}{c_{2}} ≥ (w′_{i}, w_{i−1}, …, w_{1})

for some i ≥ 1, where w′_{i} > w_{i} (note that L [figure] \binom{p_{2}}{c_{2}} cannot contract to the empty graph because τ is assumed to be minimal). Then 6.1 gives

c̃_{2} = det(L [figure] \binom{p_{2}}{c_{2}}) = det(w′_{i}, w_{i−1}, …, w_{1}) < det(w_{i}, w_{i−1}, …, w_{1})

≤ det(w_{s}, w_{s−1}, …, w_{1}) = det(G_{(−1)} [figure] \binom{p_{1}}{c_{1}}) = c_{1},

the last equality by 2.12. This proves (26).

Note that τ ∈ E implies that a_{1}Δ(τ) ≥ 2 so, by 6.2 and (26),

c_{2} = a_{1}c_{1}Δ(τ) − c̃_{2} ≥ 2c_{1} − c̃_{2} > c_{1}.

This shows that c_{1} < c_{2} whenever τ ∈ E^{+} is minimal in 𝕋(‡). In view of the first part of the proof, we obtain the “only if” part of assertion (2), i.e., τ ∈ E^{+} ⟹ c_{1} < c_{2}. The converse is much easier: If τ ∈ E \ E^{+}, applying the “only if” part of (2) to τ^{×} ∈ E^{+} gives c_{2} < c_{1}; thus (given τ ∈ E) c_{1} ≤ c_{2} ⟹ τ ∈ E^{+} and (2) holds.

If τ ∈ E is minimal in 𝕋(‡) then (6.4) τ̃ is defined and belongs to E; thus τ^{∗} = (τ̃)^{×} is defined and belongs to E. Observe that (c_{1}^{∗}, c_{2}^{∗}) = (c̃_{2}, c_{1}).

Suppose that τ ∈ E is minimal in 𝕋(‡). If τ ∈ E^{+} then (26) reads c_{1}^{∗} < c_{2}^{∗}, so τ^{∗} ∈ E^{+} by part (2). Conversely, if τ^{∗} ∈ E^{+} then part (2) gives c_{1}^{∗} < c_{2}^{∗}, or equivalently c̃_{2} < c_{1}; since c_{2} + c̃_{2} = a_{1}c_{1}Δ(τ) ≥ 2c_{1}, we get c_{1} < c_{2}, so τ ∈ E^{+} by part (2). Hence, τ ∈ E^{+} ⟺ τ^{∗} ∈ E^{+} and (3) is proved.

Assertion (4) follows immediately from 6.2.

If τ ∈ E^{+} then τ^{×} ∈ E \E^{+}, so τ^{×} is minimal in 𝕋(‡) by part (1), so (6.4) τ^{⋆} = (τ^{×})^{∼} is defined, minimal and belongs to E. Clearly, (τ^{⋆})^{∗} = τ ∈ E^{+}, so τ^{⋆} ∈ E^{+} by part (3). This shows that (5) holds and (6) is obvious. □

Corollary 6.8. For each element τ of E_{NM} = {τ ∈ E | τ is not minimal in (𝕋(‡), <)}, define [τ] = {τ, τ^{⋆}, (τ^{⋆})^{⋆}, … }. Then {[τ] | τ ∈ E_{NM}} is a partition of E^{+}. 6.9. Suppose that τ ∈ E^{+} is minimal in (𝕋(‡), <) and that, for some surface X satisfying (‡), τ ∈ 𝕋(X). Then τ^{∗} ∈ 𝕋(X). Indeed, τ̃ ≡ τ by part 2 of 6.2, so τ̃ ∈ 𝕋(X) by 2.25, and consequently τ^{∗} = (τ̃)^{×} ∈ 𝕋(X). Corollary 6.10. If X is a surface satisfying (‡) and such that 𝕋(X) ∩ E ≠ ∅, then X admits a basic affine ruling of type II.

Proof. Choose τ_{1} ∈ 𝕋(X) ∩ E; replacing τ_{1} by τ_{1}^{×} if necessary, we may arrange that τ_{1} ∈ E^{+}. Then (6.8) τ_{1} ∈ [τ] for some τ ∈ E_{NM} and, by iterating 6.9, we obtain τ ∈ 𝕋(X). Since τ is non-minimal, we may consider τ′ ∈ 𝕋(‡) such that τ > τ′; note that T′_{1} has two columns but T′_{2} has at most one, where τ′ = (m, T′_{1}, T′_{2}). We have τ′ ∈ 𝕋(X) by 2.25, so τ′ = disc(X, Λ, F) for some affine ruling Λ of X and some F ∈ Λ_{∗}. Since T′_{1} (resp. T′_{2}) has two (resp. at most one) columns, Λ is basic and of type II. □

Lemma 6.11. If a_{0}, a_{1}, a_{2} are pairwise relatively prime positive integers then

𝕋_{III}(a_{0}, a_{1}, a_{2}) ⊂ E.

Proof. Let τ = (m, T_{1}, T_{2}) ∈ 𝕋_{III}(a_{0}, a_{1}, a_{2}); by 2.28, we may write T_{i} = ( \begin{matrix}p_{i} & 1 \\ c_{i} & a_{i} \end{matrix}) (i = 1, 2). Define G_{0}, G_{1}, G_{2} as in 2.26 and let us also write Γ = G_{0}; then det(G_{i}) = a_{i} (all i = 0, 1, 2) and a calculation using 2.12 and 1.11 gives det(G_{0}) = Δ(τ)a_{1}a_{2}c_{1}c_{2} −a_{1}c_{1}^{2} − a_{2}c_{2}^{2}.

By 2.12, G_{1} has discriminant a_{1} and subdiscriminants a_{1} − 1 and a_{1} − 1. Since τ ∈ 𝕋_{III}(a_{0}, a_{1}, a_{2}), this implies that (−1)a_{2} ≡ a_{0} (mod a_{1}), so a_{0} + a_{1} + a_{2} ≡ 0 (mod a_{1}). Similarly, a_{0} + a_{1} + a_{2} ≡ 0 (mod a_{2}). Since gcd(a_{1}, a_{2}) = 1, this implies

(28)  a_{0} + a_{1} + a_{2} = γa_{1}a_{2}, for some γ ∈ ℤ, γ ≥ 1.

Note that G_{0} shrinks to an admissible chain and has discriminant a_{0}. Since (γa_{2} − 1)a_{1} ≡ a_{2} and (γa_{1} − 1)a_{2} ≡ a_{1} (mod a_{0}), the fact that τ belongs to 𝕋_{III}(a_{0}, a_{1}, a_{2}) implies that the subdiscriminants of G_{0} are congruent to γa_{1} − 1 and γa_{2} − 1 modulo a_{0}. On the other hand, the linear chain

[figure: Γ′:; −a_{1}; −γ; −a_{2}]

shrinks to an admissible chain, has discriminant a_{0} and subdiscriminants γa_{1} − 1 and γa_{2} − 1. So, by 1.15,

(29)  G_{0} is equivalent to Γ′.

In order to show that τ ∈ E, there remains to show that Δ(τ) ≠ 1 or min(a_{1}, a_{2}) ≠ 1. Assume the contrary: Δ(τ) = 1 and min(a_{1}, a_{2}) = 1. Replacing τ by τ^{×} if necessary, we will assume from now-on:

6.11.1. e_{1} disappears before e_{2}.

(By 6.5, at least one of e_{1}, e_{2} disappears in the shrinking process which transforms Γ = G_{0} into an admissible chain—note that 6.5 is valid whenever τ satisfies conditions (1) and (2) of 6.3, which is the case here.) In particular we have m = 1, since G_{0} is not a minimal weighted tree.

We will obtain a contradiction only after having established several facts. We begin with:

6.11.2. a_{1} = 1, a_{2} ≥ 5, c_{1} > c_{2} and the contraction of Δ̃ increases the weight of e_{2} by more than 1.

To see this, consider the result of shrinking Δ̃ in (23) (where Γ = G_{0}):

(30)  [figure: …; …; e_{2}; y; e_{1}; x]

Since e_{1} disappears before e_{2}, we must have x = −1 and y < x; thus −1−a_{2} < y < −1, so a_{2} ≥ 2 and consequently a_{1} = 1. Let us be more precise. Since, in (23), Δ̃ contains at least 3 vertices, we may consider the situation where there remains two vertices in Δ̃:

[figure: …; …; e_{2}; e_{1}; w_{2}; x_{2}; x_{1}; w_{1}]

(where w_{1} ≥ −2, since a_{1} = 1). Since this contracts to (30), we must have (x_{1}, x_{2}) = (−1, −2) or (−2, −1); in fact we must have (x_{1}, x_{2}) = (−2, −1) because the other possibility would give x ≥ 0 in (30), which is absurd. So the contraction of Δ̃ increases the weight of e_{2} by more than 1. Recall that a_{0} = Δ(τ)a_{1}a_{2}c_{1}c_{2} − a_{1}c_{1}^{2} − a_{2}c_{2}^{2} is strictly positive; with Δ(τ) = 1 = a_{1}, this implies that a_{2}c_{2}(c_{1} − c_{2}) > c_{1}^{2}, so c_{1} > c_{2} and a_{2} ≥ 5, which proves 6.11.2.

6.11.3. G_{0} is equivalent to a tree with two vertices, one of which has weight −a_{2}. Moreover, if Γ′′ is any tree with two vertices and equivalent to G_{0}, then one of the weights in Γ′′ is −a_{2}.

The first assertion is (29) with a_{1} = 1; the second sentence follows easily from the first one. We also claim:

6.11.4. τ is minimal in 𝕋(‡).

Assume the contrary then, arguing as in the proof of 6.7 (see (24) and (25)), we obtain

T_{1} = ( \begin{matrix}1 & 1 \\ c_{1} & 1 \end{matrix} ) and T_{2} = ( \begin{matrix}c_{1}−2 & 1 \\ c_{1}−1 & a_{2} \end{matrix} ). Then G_{0} = (−c_{1} + 1, −1 − a_{2}, [c_{1} − 2], −1, −c_{1}, [c_{1}]) ≥

(−c_{1} + 1, −a_{2} + c_{1} + 1); since −a_{2} + c_{1} + 1 ≠ −a_{2}, 6.11.3 implies that −c_{1} + 1 = −a_{2}, so the other weight is −a_{2} + c_{1} + 1 = 2 ≥ 0, which is absurd.

Recall that m = 1 and let us use the notation:

(31)  L = G_{(−1)} [figure] T_{1} = (0, −1, [n_{0}], z_{1}, …, [n_{h−1}], z_{h}, [n_{h}])

where n_{j} ≥ 0, z_{j} ≤ −3 and where the distinguished vertex is the one of weight 0. Note that the hypothesis of 6.2 is satisfied, so \binom{p_{2}}{c_{2}} ∈ Cont(L) and τ̃ is defined. In particular, Cont(L) contains a nonempty tableau, so h ≥ 1.

6.11.5. h ≥ 2 and, for some i ∈ {1, …, h − 1},

(32)  G_{0} = L [figure] T_{2} ≥ (…, z_{i−1}, [n_{i−1}], z_{i} + 1, −a_{2}, −1, [n_{i}], z_{i+1}, …).

Moreover, n_{i} ≥ 2 and e_{1} is either the leftmost or the rightmost vertex in [n_{i}].

We have \binom{p_{2}}{c_{2}} = cont(L, i; x, y) for some i ∈ \{1, …, h\} (for suitable x, y); then

(33)  L\binom{p_{2}}{c_{2}} ≥ (…, z_{i−1}, [n_{i−1}], x, 0^{∗}, y, [n_{i}], z_{i+1}, …),

or equivalently:

(34)  G_{0} = L [figure] T_{2} ≥ (…, z_{i−1}, [n_{i−1}], x, −a_{2}, y, [n_{i}], z_{i+1}, …),

where e_{2} is the vertex of weight −a_{2}. Since the contraction (34) increases the weight of e_{2} by only 1, 6.11.2 implies that some vertex of Δ̃ is still present in the right hand side of (34). It follows that the vertex of weight y belongs to Δ̃, so x = z_{i} + 1, y = −1 and (32) holds. Since e_{1} disappears before e_{2}, e_{1} is in [n_{i}]. If i = h then the right hand side of (33) shrinks to a linear weighted pair, which contradicts 6.11.4 (2.15, 2.19); so i < h and consequently h ≥ 2. Since c_{1} > c_{2}, we have in particular c_{1} > 2; so e_{1} has two neighbors in L, one of them has weight −2 and the other has weight strictly less than −2. This proves 6.11.5.

Obseve that τ̃ is defined and satisfies the hypothesis of the Lemma as well as Δ(τ̃) = 1 = a_{1}. We claim that τ̃ also satisfies 6.11.1: if not, then (τ̃)^{×} does, so 6.11.2 applied to (τ̃)^{×} gives c̃_{2} > c_{1}, which is not the case because we have c_{2} + c̃_{2} = a_{1}c_{1}Δ(τ) = c_{1}, so c_{1} > c̃_{2}. So we may, if we want, replace τ by τ̃. Note, however, that if (in 6.11.5) e_{1} is the leftmost vertex of [n_{i}], then the contrary claim holds for τ̃. In other words, we may arrange that:

6.11.6. e_{1} is the rightmost vertex of [n_{i}].

Consider the weighted pair Z consisting of a single vertex of weight 0; then we may write Z\binom{p_{1}}{c_{1}} in one of the following forms:

(a) ([x_{h}], y_{h−1}, …, y_{4}, [x_{3}], y_{2}, [x_{1}], −1^{∗}, y_{1}, [x_{2}], y_{3}, [x_{4}] …, [x_{h−1}], y_{h}),

(b) (y_{h}, [x_{h−1}], …, y_{4}, [x_{3}], y_{2}, [x_{1}], −1^{∗}, y_{1}, [x_{2}], y_{3}, [x_{4}] …, y_{h−1}, [x_{h}]), where y_{j} ≤ −3, x_{j} ≥ 0 and x_{h} > 0; e_{1} is the vertex of weight −1 and the unique vertex of Z is the leftmost vertex in (a) or (b). Note that, because of 6.11.6, we don’t need to consider more cases than (a) and (b) (i.e., cases of the type (…, y_{1}, −1^{∗}, [x_{1}], …)); note, also, that h is odd in case (a) and even in case (b). The fact that (a) (resp. (b)) shrinks to a single vertex of weight 0 gives:

(35)  x_{j} + y_{j} = \{ \begin{matrix}−3, & if 1 < j < h, \\ −2, & if j = 1 or j = h. \end{matrix}

Note that z_{i} = y_{2}, n_{i} = x_{1} + 1, z_{i+1} = y_{1}, etc., and rewrite (32) as (36)

G_{0} ≥ \{ \begin{matrix}([x_{h}], y_{h−1}, …, [x_{3}], y_{2} + 1, −a_{2}, −1, [x_{1}], −2, y_{1}, [x_{2}], …, [x_{h−1}], y_{h}), & or \\ (y_{h}, [x_{h−1}], …, [x_{3}], y_{2} + 1, −a_{2}, −1, [x_{1}], −2, y_{1}, [x_{2}], …, y_{h−1}, [x_{h}]), \end{matrix}

in cases (a) and (b) respectively. Next we show:

6.11.7. Case (a) is impossible.

Assume that we are in case (a). By (36),

(37)  G_{0} ≥ ([x_{h}], y_{h−1}, …, [x_{3}], y_{2} + 1, −a_{2} + x_{1} + 2, y_{1} + 1, [x_{2}], …, [x_{h−1}], y_{h}),

where the right hand side contains at least 5 vertices (h ≥ 2 by 6.11.5, so h ≥ 3 since it is odd; also recall that x_{h} > 0). By 6.11.3, −a_{2} + x_{1} + 2 = −1, so:

(38)  a_{2} = x_{1} + 3;

together with (37), this gives

(39)  G_{0} ≥ ([x_{h}], y_{h−1}, …, [x_{3}], y_{2} + 2, y_{1} + 2, [x_{2}], …, [x_{h−1}], y_{h}),

which has at least 4 vertices. So −1 ∈ {y_{2} + 2, y_{1} + 2}. If y_{1} + 2 = −1 then the right hand side of (39) shrinks to (…, [x_{3}], y_{2} + 2 + x_{2} + 1, …) = (…, [x_{3}], 0, …) by (35); since there can’t be a nonnegative weight in a tree equivalent to G_{0}, we conclude that y_{2} + 2 = −1 and, by (39),

(40)  G_{0} ≥ ([x_{h}], y_{h−1}, …, y_{4} + 1, x_{3} + y_{1} + 3, [x_{2}], …, [x_{h−1}], y_{h}).

Note that if h = 3 then (40) reads G_{0} ≥ (x_{3} + y_{1} + 3, [x_{2}], y_{3}). More generally, we claim:

(41)  G_{0} ≥ (x_{h} + y_{h−2} + p, [x_{h−1}], y_{h}), where p = \{ \begin{matrix}3, & if h = 3, \\ 2, & if h > 3. \end{matrix}

Indeed, if h > 3 then we can continue contracting (40) as long as we have more than 2 vertices. At each stage of the process, the next vertex to disappear is clearly identified and the contraction process inescapably leads to the right hand side of (41), unless contraction stops before that point; since the right hand side of (41) has at least 2 vertices, contraction doesn’t stop before that point and (41) holds. Now (41) implies that, if h > 3,

x_{3} + y_{1} = −4,  x_{2} + y_{4} = −3,
(42)  x_{5} + y_{3} = −3,  x_{4} + y_{6} = −3,
⋮  ⋮
x_{h−2} + y_{h−4} = −3, x_{h−3} + y_{h−1} = −3.

(These are obtained by writing down, at each stage of the contraction process, the equation which corresponds to the fact that the next vertex to disappear has weight −1.) Since G_{0} contracts to an admissible chain, (41) implies that x_{h} + y_{h−2} + p ≤ −1, so x_{h} + y_{h−2} ≤ −3; together with the first column of (42), this gives:

(43)  x_{j} + y_{j−2} ≤ −3, for all odd j such that 3 ≤ j ≤ h.

(Note that, although the notation in (42) assumes that h > 3, (43) is valid when h = 3 as well.) We claim that:

(44)  y_{j} > −a_{2} for all odd j such that 1 ≤ j ≤ h.

Indeed, y_{1} = −2 − x_{1} > −3 − x_{1} = −a_{2}, by (35) and (38); if j > 1 then y_{j} ≥ −3 − x_{j} ≥ y_{j−2}, by (35) and (43), so (44) holds.

We may now obtain a contradiction from 6.11.3, (41) and (44): If x_{h} +y_{h−2} +p = −1 then G_{0} ≥ (−1, y_{h}) by (41), so y_{h} = −a_{2} by 6.11.3, and this contradicts (44). If x_{h} + y_{h−2} + p < −1, then the right hand side of (41) must be an admissible chain with exactly two vertices (x_{h−1} = 0); since y_{h} ≠ −a_{2} by (44), we have x_{h} + y_{h−2} + p = −a_{2} by 6.11.3, but this is absurd because y_{h−2} > −a_{2} and x_{h} + p > 0. This proves 6.11.7.

6.11.8. Case (b) is impossible.

This is very similar to 6.11.7 and we only sketch the argument. Assume that we are in case (b) (so h is even). By (36),

(37’)  G_{0} ≥ (y_{h}, [x_{h−1}], …, [x_{3}], y_{2} + 1, −a_{2} + x_{1} + 2, y_{1} + 1, [x_{2}], …, y_{h−1}, [x_{h}])

and we deduce that a_{2} = x_{1} + 3 and y_{2} + 2 = −1 (as before); we also find:

(41’)  G_{0} ≥ (y_{h−1} + p, [x_{h}]), where p = \{ \begin{matrix}3, & if h = 2, \\ 2, & if h > 2, \end{matrix}

and if h > 2:

x_{3} + y_{1} = −4,  x_{2} + y_{4} = −3,
(42’)  x_{5} + y_{3} = −3,  x_{4} + y_{6} = −3,
⋮  ⋮
x_{h−1} + y_{h−3} = −3, x_{h−2} + y_{h} = −3.

Then (42’) implies (43), and (44) follows; together with 6.11.3 and (41’), this gives a contradiction. So 6.11.8 holds and the Lemma is proved. □

Corollary 6.12. Let X be a surface of type [a, b, c] for some pairwise relatively prime integers a, b, c ≥ 1. Then X is isomorphic to ℙ(a, b, c).

Proof. By 2.22.2, X admits a basic affine ruling Λ; if Λ is of type I or II then the assertion follows from 3.2 and 5.2.

Suppose that Λ is of type III, choose F ∈ Λ_{∗} and let τ = disc(X, Λ, F); by 2.27, τ ∈ 𝕋_{III}(a_{0}, a_{1}, a_{2}) for some permutation a_{0}, a_{1}, a_{2} of a, b, c. Then 6.11 gives τ ∈ E and, by 6.10, X admits a basic affine ruling of type II. □

Remark. Suppose X satisfies (‡). Then X has at most three singular points (1.8) and X admits a basic affine ruling (2.22.2). If X admits a basic ruling of type III (resp. II), then X has at most one (resp. two) singular points not a rational double point. Hence in case X has three singularities that are not rational double points, 3.2 gives a stronger statement than 6.12, namely:

Let X be a surface satisfying (‡). If the discriminants a_{0}, a_{1}, a_{2} of its singular points are pairwise relatively prime, and if X has three singularities that are not rational double points, then X = ℙ(a_{0}, a_{1}, a_{2}) and no a_{i} divides the sum of the other two.

Corollary 6.13. Let a_{0}, a_{1}, a_{2} be pairwise relatively prime positive integers. Then the set 𝕋_{III}(a_{0}, a_{1}, a_{2}) ∪ 𝕋_{III}(a_{0}, a_{2}, a_{1}) is nonempty if and only if a_{1}a_{2} | a_{0} + a_{1} + a_{2}. Moreover, 𝕋_{III}(a_{0}, a_{1}, a_{2}) ∪ 𝕋_{III}(a_{0}, a_{2}, a_{1}) is equal to:

⋃_{τ∈E}{τ, τ^{⋆}, (τ^{⋆})^{⋆}, ((τ^{⋆})^{⋆})^{⋆}, … } ∪ {τ^{×}, (τ^{⋆})^{×}, ((τ^{⋆})^{⋆})^{×}, (((τ^{⋆})^{⋆})^{⋆})^{×}, … }

where E denotes the set of elements of 𝕋_{III}(a_{0}, a_{1}, a_{2}) ∪ 𝕋_{III}(a_{0}, a_{2}, a_{1}) which are nonminimal in 𝕋(‡).

Proof. Follows from 6.11 and 6.8. □

7. Explicit description of the set 𝕋_{0}(ℙ)

Let ℙ = ℙ(a, b, c), where a, b, c are pairwise relatively prime positive integers. By [2], it is clear that the problem of describing all affine rulings of ℙ reduces to that of describing the set 𝕋_{0}(ℙ). Now we have: Corollary 7.1. 𝕋_{0}(ℙ) is the union of the sets 𝕋_{P}(a_{0}, a_{1}, a_{2}), for all P ∈ {I, II.1, II.2, III} and all permutations (a_{0}, a_{1}, a_{2}) of (a, b, c).

Proof. By 2.27, 𝕋_{0}(ℙ) ⊆ ⋃ 𝕋_{P}(a_{0}, a_{1}, a_{2}). For the reverse inclusion, consider τ = (m, T_{1}, T_{2}) ∈ 𝕋_{P}(a_{0}, a_{1}, a_{2}); then τ = disc[X, Λ, F] for some [X, Λ, F] ∈ 𝕊_{0}(‡), because disc : 𝕊_{0}(‡) → 𝕋_{0}(‡) is surjective (2.24). Then (2.23) the resolution graph of X is equivalent to (G_{(−m)} ⊖ T_{1}) ⊖ T_{2}, which is equivalent to G_{[a_{0},a_{1},a_{2}]} = G_{[a,b,c]} by definition of 𝕋_{P}(a_{0}, a_{1}, a_{2}). So X is a surface of type [a, b, c] and 6.12 implies that X = ℙ. Consequently, τ ∈ 𝕋_{0}(ℙ). □

So our task is to describe the set 𝕋_{P}(a_{0}, a_{1}, a_{2}) explicitely, for each permutation (a_{0}, a_{1}, a_{2}) of (a, b, c) and each P ∈ {I, II.1, II.2, III}. We begin with an observation: 7.2. 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. Proposition 7.3. Given pairwise relatively prime positive integers a_{0}, a_{1}, a_{2}, 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}).

Proof. Clear from the proof of 3.2. □

Proposition 7.4. Given pairwise relatively prime positive integers a_{0}, a_{1}, a_{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 7.2 and define

(45)  \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

(46)  (1, \binom{x′_{1}}{a_{1}}, (\begin{matrix}p_{2} & 1 \\ c_{2} & a_{2} \end{matrix})) .

Remark. Since τ ↦ τ^{×} is a bijection 𝕋_{II.1}(a_{0}, a_{2}, a_{1}) → 𝕋_{II.2}(a_{0}, a_{1}, a_{2}), a description of 𝕋_{II.2}(a_{0}, a_{1}, a_{2}) is easily obtained from the above statement.

Proof. Suppose that τ = (1, T_{1}, T_{2}) ∈ 𝕋_{II.1}(a_{0}, a_{1}, a_{2}) and write T_{1} = ( \begin{matrix}p \\ a_{1} \end{matrix} ) and T_{2} = ( \begin{matrix}p_{2} & 1 \\ c_{2} & a_{2} \end{matrix} ) (see 2.28). We saw, at the beginning of the proof of 5.1, that a_{2} | a_{0} + a_{1}; so (a_{0} + a_{1} + a_{2})/a_{2} is a natural number at least 2 (we will see, below, that it is greater than 2). In particular, we have x_{2} ∈ {0, 1} by 7.2.

By 5.1, τ is not minimal and its immediate predecessor τ′ belongs to 𝕋_{I}(a_{0}, a_{1}, a_{2}), so (by 7.3) τ′ = (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}). This implies that τ = (1, \binom{x′_{1}}{a_{1}}, T \binom{x_{2}}{a_{2}}), where x′_{1} is defined in 7.2 and T ∈ T_{x_{0}−1}(L^{t}), with L = G_{(−1)} [figure] \binom{x_{1}}{a_{1}}. Note that the first column of T must be \binom{p_{2}}{c_{2}} and that we may write T = \binom{p_{2}}{c_{2}}\binom{1}{1}^{1−x_{2}} (recall that x_{2} ∈ \{0, 1\}). So \binom{p_{2}}{c_{2}}\binom{1}{1}^{1−x_{2}} ∈ T_{x_{0}−1}(L^{t}), which implies \binom{p_{2}}{c_{2}} ∈ T_{x_{0}−x_{2}}(L^{t}); thus \binom{p_{2}}{c_{2}} is the matrix product M(L^{t})\binom{1}{x_{0}−x_{2}}, which is the

same as M(L)^{t}\binom{1}{x_{0}−x_{2}} by 2.14. By 5.38 of [2] we have M(L) = ( \begin{matrix}a_{1}−x_{1}−x′_{1}+x′′_{1} & a_{1}−x_{1} \\ a_{1}−x′_{1} & a_{1} \end{matrix} ), so

(45) and (46) hold.

If (a_{0} + a_{1} + a_{2})/a_{2} = 2 then a_{0} + a_{1} = a_{2}; feeding this in Eq(a_{0}, a_{1}, a_{2}) and manipulating gives x_{1} = a_{1} − 1 = x′_{1}, x′′_{1} = a_{2} − 2 and x_{0} = x_{2}; then (45) gives p_{2} = 0, which is absurd. Hence, (a_{0} + a_{1} + a_{2})/a_{2} > 2.

Conversely, suppose that (a_{0} + a_{1} + a_{2})/a_{2} is a natural number greater than 2; in order to show that 𝕋_{II.1}(a_{0}, a_{1}, a_{2}) is nonempty, consider the unique element τ′ = (x_{0}, \binom{x_{1}}{a_{1}}, \binom{x_{2}}{a_{2}}) of 𝕋_{I}(a_{0}, a_{1}, a_{2}), let L = G_{(−1)} [figure] \binom{x_{1}}{a_{1}} and define \binom{p_{2}}{c_{2}} = M(L^{t})\binom{1}{x_{0}−x_{2}}. Note that x_{0} > 0 and x_{2} ∈ {0, 1} imply x_{0} − x_{2} ≥ 0. We claim:

(47)  \binom{p_{2}}{c_{2}} ∈ T_{x_{0}−x_{2}}(L^{t}).

If this is the case then it is easy to see that we may construct an element τ of 𝕋_{II.1}(a_{0}, a_{1}, a_{2}) by reading the above argument backward. Observe that, by definition of T_{x_{0}−x_{2}}(L^{t}), if (47) is false then we must have x_{0} − x_{2} = 0, so (i) x_{0} = 1 = x_{2}; and (ii) L must satisfy the condition “w_{i} = −2 for all i” (see 2.13). Now (i) and Eq(a_{0}, a_{1}, a_{2}) give a_{0} + a_{1} = a_{2}(a_{1} − x_{1}) and (ii) gives x_{1} = a_{1} − 1, so a_{0} + a_{1} = a_{2}, a contradiction. So (47) holds and the proof is complete. □

Proposition 7.5. Let a_{0}, a_{1}, a_{2} be pairwise relatively prime positive integers. Then at most one element of 𝕋_{III}(a_{0}, a_{1}, a_{2}) is non-minimal in 𝕋(‡) and such an element exists if and only if a_{1}a_{2} | a_{0} + a_{1} + a_{2} and a_{0} > a_{1} − a_{2}. Moreover, if such an element τ exists then τ = (1, ( \begin{matrix}p_{1} & 1 \\ c_{1} & a_{1} \end{matrix} ) , ( \begin{matrix}p_{2} & 1 \\ c_{2} & a_{2} \end{matrix} )), where

(48)  \binom{p_{1}}{c_{1}} = \binom{1}{a_{2} − x_{2}} + (x_{0} − x_{1})\binom{x_{2}}{a_{2}},
(49)  \binom{p_{2}}{c_{2}} = \binom{a_{1}p_{1}(c_{1} − p_{1}) − 1}{a_{1}c_{1}p_{1} − 1} + (1 − x_{2})\binom{a_{1}c_{1}(c_{1} − p_{1}) − 1}{a_{1}c_{1}^{2}}

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

Proof. Suppose that τ ∈ 𝕋_{III}(a_{0}, a_{1}, a_{2}) is a non-minimal element of 𝕋(‡). By 2.31, the immediate predecessor τ_{1} of τ belongs to 𝕋_{II.2}(a_{0}, a_{1}, a_{2}), so τ_{1}^{×} ∈ 𝕋_{II.1}(a_{0}, a_{2}, a_{1}). Now 7.4 describes τ_{1}^{×} as follows: Let (x_{0}, x_{2}, x_{1}) be the unique solution to Eq(a_{0}, a_{2}, a_{1}) (equivalently, (x_{0}, x_{1}, x_{2}) solves Eq(a_{0}, a_{1}, a_{2})) and define x′_{2}, x′′_{2} as in 7.2. By (28), a_{1}a_{2} divides a_{0} + a_{1} + a_{2}; thus x_{1}, x_{2} ∈ {0, 1}, x′_{2} = x_{2} and x′′_{2} = x_{2} − 1. (Note, also, that 𝕋_{II.1}(a_{0}, a_{2}, a_{1}) ≠ ∅ implies that (a_{0} + a_{1} + a_{2})/a_{1} > 2, so a_{0} > a_{1} − a_{2}.) Define

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

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

then (7.4) τ_{1}^{×} = (1, \binom{x′_{2}}{a_{2}}, ( \begin{matrix}p′_{1} & 1 \\ c_{1} & a_{1} \end{matrix} )) = (1, \binom{x_{2}}{a_{2}}, ( \begin{matrix}p′_{1} & 1 \\ c_{1} & a_{1} \end{matrix} )), so τ_{1} = (1, ( \begin{matrix}p′_{1} & 1 \\ c_{1} & a_{1} \end{matrix} ) , \binom{x_{2}}{a_{2}}).

Let L = G_{(−1)} [figure] ( \begin{matrix}p′_{1} & 1 \\ c_{1} & a_{1} \end{matrix} ). Then τ = (1, ( \begin{matrix}c_{1}−p′_{1} & 1 \\ c_{1} & a_{1} \end{matrix} ) , T \binom{x_{2}}{a_{2}}), where T ∈ T_{0}(L^{t}). Let \binom{p_{2}}{c_{2}}

be the first column of T, then \binom{p_{2}}{c_{2}}\binom{1}{1}^{1−x_{2}} = T ∈ T_{0}(L^{t}), so \binom{p_{2}}{c_{2}} ∈ T_{1−x_{2}}(L^{t}), so \binom{p_{2}}{c_{2}} =

M(L^{t})\binom{1}{1−x_{2}} = M(L)^{t}\binom{1}{1−x_{2}}. Now 5.38 of [2] gives M(L) = ( \begin{matrix}a_{1}p′_{1}(c_{1}−p′_{1})−1 & a_{1}c_{1}^{2}−a_{1}c_{1}p′_{1}−1 \\ a_{1}c_{1}p′_{1}−1 & a_{1}c_{1}^{2} \end{matrix} ),

so

(51)  \binom{p_{2}}{c_{2}} = \binom{a_{1}p′_{1}(c_{1} − p′_{1}) − 1}{a_{1}c_{1}^{2} − a_{1}c_{1}p′_{1} − 1} + (1 − x_{2})\binom{a_{1}c_{1}p′_{1} − 1}{a_{1}c_{1}^{2}}.

Now τ = (1, ( \begin{matrix}c_{1}−p′_{1} & 1 \\ c_{1} & a_{1} \end{matrix} ) , ( \begin{matrix}p_{2} & 1 \\ c_{2} & a_{2} \end{matrix} )) = (1, ( \begin{matrix}p_{1} & 1 \\ c_{1} & a_{1} \end{matrix} ) , ( \begin{matrix}p_{2} & 1 \\ c_{2} & a_{2} \end{matrix} )), where p_{1} = c_{1} − p′_{1}. Formulas

(48) and (49) are obtained from (50), (51) and p_{1} = c_{1} − p′_{1}.

We leave it to the reader to verify that, 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}) contains a non-minimal element of 𝕋(‡) (there is a similar argument in the proof of 7.4). □

Our next task is to make 6.13 more explicit; this is done in 7.7, below.

Definition 7.6. 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.

7.6.1. 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}.

7.6.2. Let M = ( \begin{matrix}1 & 1 \\ 1 & 1 \end{matrix} ) and define u_{n} = s_{n}(M) and v_{n} = t_{n}(M). Note that the beginning terms of these two sequences are:

\begin{matrix}n & 0 & 1 & 2 & 3 & 4 & … \\ u_{n} & 1 & 1 & a_{2}γ − 1 & a_{2}γ(a_{1}γ − 1) − 1 & a_{2}γ[a_{1}γ(a_{2}γ − 1) − 1] − (a_{2}γ − 1) & … \\ v_{n} & 1 & 1 & a_{1}γ − 1 & a_{1}γ(a_{2}γ − 1) − 1 & a_{1}γ[a_{2}γ(a_{1}γ − 1) − 1] − (a_{1}γ − 1) & … \end{matrix}

7.6.3. 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′). Note that, in each of the following cases:

(i) a_{1} > 1 and a_{2} > 1; (ii) 1 = a_{1} < a_{2}; (iii) a_{1} > a_{2} = 1; (iv) a_{1} = 1 = a_{2},

the beginning terms of {ξ_{n}}_{n=0}^{∞} and {η_{n}}_{n=0}^{∞} are as follows:

n  0  1 2  3  4  …
(i)  ξ_{n}  −γ  0 γ  a_{2}γ^{2}  a_{1}a_{2}γ^{3} − γ  …
η_{n}  −γ  0 γ  a_{1}γ^{2}  a_{1}a_{2}γ^{3} − γ  …
(ii)  ξ_{n}  −γ  −1 γ a_{2}γ + 1  a_{2}γ^{3} − γ  …
η_{n}  −γ − 1 0 1  γ^{2}  γ(a_{2}γ + 1) − 1 …
(iii)  ξ_{n}  −γ − 1 0 1  γ^{2}  γ(a_{1}γ + 1) − 1 …
η_{n}  −γ  −1 γ a_{1}γ + 1  a_{1}γ^{3} − γ  …
(iv) ξ_{n} = η_{n} −γ − 1 −1 1 γ + 1  γ(γ + 1) − 1 …

7.6.4. 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} ))

(we are not claiming that these always belong to 𝕋(‡)). 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}

Proposition 7.7. 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}, … }.

Observe, also, that 𝕋_{III}(a_{0}, a_{2}, a_{1}) = {τ^{×} | τ ∈ 𝕋_{III}(a_{0}, a_{1}, a_{2})} holds in all cases.

Proof of 7.7. The fact that 𝕋_{III}(a_{0}, a_{1}, a_{2}) is nonempty if and only if a_{1}a_{2} divides a_{0} + a_{1} + a_{2} is an immediate consequence of 6.13. Assume that a_{1}a_{2} | a_{0} + a_{1} + a_{2}. If a_{0} > a_{1} − a_{2} then, by 7.5, 𝕋_{III}(a_{0}, a_{1}, a_{2}) has a unique element which is non-minimal in 𝕋(‡), and a direct calculation shows that this element is f_{2} (one verifies that, in each of the cases (i–iv) of 7.6.3, the element τ given by 7.5 is equal to f_{2}). Similarly, if a_{0} > a_{2} −a_{1} then the unique element of 𝕋_{III}(a_{0}, a_{2}, a_{1}) which is non-minimal in 𝕋(‡) can be seen to be g_{2}^{×}. Again by calculation, one checks that {f_{2}, f_{2}^{⋆}, (f_{2}^{⋆})^{⋆}, … } = {f_{2}, g_{3}^{×}, f_{4}, g_{5}^{×}, … } and that {g_{2}^{×}, (g_{2}^{×})^{⋆}, ((g_{2}^{×})^{⋆})^{⋆}, … } = {g_{2}^{×}, f_{3}, g_{4}^{×}, f_{5}, … } (one can use parts (4) and (5) of 6.2 to compute τ ↦ τ^{⋆} explicitely). The desired result follows from this and 6.13. □

Example 7.8. The following is a description of 𝕋_{0}(ℙ^{2}). First, 7.3 and 7.4 give: • 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 7.7; by 7.6.3 (case (iv), 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} )) , … \} .

Example 7.9. We now describe 𝕋_{0}(ℙ(2, 3, 5)). By 7.3 and 7.4, • 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 7.7; by 7.6.3 (case (i), 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) = {g_{2}^{×}, f_{3}^{×}, g_{4}^{×}, f_{5}^{×}, … }

= \{(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} )) , … \} .

8. Further remarks

Corollary 8.1. Let a_{0}, a_{1}, a_{2} be pairwise relatively prime positive integers. Then

Δ(τ) = \frac{a_{0} + a_{1} + a_{2}}{a_{1}a_{2}} , for all τ ∈ 𝕋_{III}(a_{0}, a_{1}, a_{2}) ∪ 𝕋_{III}(a_{0}, a_{2}, a_{1}).

Proof. By 6.11, 6.7 and 6.8, there exists τ′ ∈ E_{NM} ∩ 𝕋_{III}(a_{0}, a_{i}, a_{j}) such that Δ(τ′) = Δ(τ) (for a suitable permutation i, j of 1, 2); thus we may assume that τ ∈ 𝕋_{III}(a_{0}, a_{1}, a_{2}) is non-minimal in 𝕋(‡). Write τ = (1, ( \begin{matrix}p_{1} & 1 \\ c_{1} & a_{1} \end{matrix} ) , ( \begin{matrix}p_{2} & 1 \\ c_{2} & a_{2} \end{matrix} )), where \binom{p_{1}}{c_{1}} and \binom{p_{2}}{c_{2}} are given

by 7.5. Then (by (49)) Δ(τ) = c_{1}c_{2} − c_{1}p_{2} − c_{2}p_{1} = | \begin{matrix}c_{1}−p_{1} & p_{2} \\ c_{1} & c_{2} \end{matrix} | = A + (1 − x_{2})B,

where A = | \begin{matrix}c_{1}−p_{1} & a_{1}p_{1}(c_{1}−p_{1})−1 \\ c_{1} & a_{1}c_{1}p_{1}−1 \end{matrix} | = p_{1} and B = | \begin{matrix}c_{1}−p_{1} & a_{1}c_{1}(c_{1}−p_{1})−1 \\ c_{1} & a_{1}c_{1}^{2} \end{matrix} | = c_{1}. Thus Δ(τ) =

p_{1} + (1 − x_{2})c_{1} = 1 + (x_{0} − x_{1})x_{2} + (1 − x_{2})(a_{2} − x_{2} + (x_{0} − x_{1})a_{2}) = x_{0} − x_{1} − x_{2} + 2, where the second equality follows from (48) and the third equality can be verified in each of the cases: x_{2} = 0, x_{2} = 1. On the other hand, Eq(a_{0}, a_{1}, a_{2}) gives

\frac{a_{0} + a_{1} + a_{2}}{a_{1}a_{2}} = x_{0} + \frac{1 − x_{1}}{a_{1}} + \frac{1 − x_{2}}{a_{2}} = x_{0} + (1 − x_{1}) + (1 − x_{2}) = Δ(τ).

□

Corollary 8.2. 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 the elements of 𝕋_{III}(a_{0}, a_{1}, a_{2}) are the triples τ = (1, ( \begin{matrix}p_{1} & 1 \\ c_{1} & a_{1} \end{matrix} ) , ( \begin{matrix}p_{2} & 1 \\ c_{2} & a_{2} \end{matrix} )) such that c_{1}, c_{2}, p_{1}, p_{2} are positive integers satisfying

γa_{1}a_{2}c_{1}c_{2} − a_{1}c_{1}^{2} − a_{2}c_{2}^{2} = a_{0}, (52) c_{1}c_{2} − c_{1}p_{2} − c_{2}p_{1} = γ (0 < p_{i} < c_{i}, i = 1, 2). (53)

Proof. If τ = (1, ( \begin{matrix}p_{1} & 1 \\ c_{1} & a_{1} \end{matrix} ) , ( \begin{matrix}p_{2} & 1 \\ c_{2} & a_{2} \end{matrix} )) ∈ 𝕋_{III}(a_{0}, a_{1}, a_{2}) ∪ 𝕋_{III}(a_{0}, a_{2}, a_{1}) then, as noted at the beginning of the proof of 6.11, we have a_{0} = Δ(τ)a_{1}a_{2}c_{1}c_{2} − a_{1}c_{1}^{2} − a_{2}c_{2}^{2}; by 8.1, we get that (52) and (53) hold.

For the converse, we use the notations of 7.6. Observe: (i) The set of positive solutions (c_{1}, c_{2}) to (52) is {(u_{n}, v_{n+1}) | n ∈ ℕ} ∪ {(u_{n+1}, v_{n}) | n ∈ ℕ}; it follows that gcd(c_{1}, c_{2}) = 1 and consequently: (ii) If we give ourselves a solution (c_{1}, c_{2}) of (52), then (53) has at most one solution (p_{1}, p_{2}). (We leave (i) as an exercise for the reader; (ii) is obvious.)

Let τ = (1, ( \begin{matrix}p_{1} & 1 \\ c_{1} & a_{1} \end{matrix} ) , ( \begin{matrix}p_{2} & 1 \\ c_{2} & a_{2} \end{matrix} )) be such that (52) and (53) hold; by observation (i),

(54)  (c_{1}, c_{2}) = (u_{n}, v_{n+1}) or (u_{n+1}, v_{n})

for some n ≥ 0. Let n be minimal such that (54) holds and note that n ≥ 2 because (53) implies c_{1} > 1 and c_{2} > 1. Define τ′ = f_{n} if (c_{1}, c_{2}) = (u_{n}, v_{n+1}) and τ′ = g_{n} otherwise. Note that if

(55)  τ′ ∈ W_{(a_{0},a_{1},a_{2})} ∪ W^{(a_{0},a_{1},a_{2})}

holds then τ′ ∈ 𝕋_{III}(a_{0}, a_{1}, a_{2}) by 7.7, so (52) and (53) hold for τ′ by the first part of the proof, so observation (ii) implies that τ = τ′ ∈ 𝕋_{III}(a_{0}, a_{1}, a_{2}) and we are done.

Since τ′ is f_{n} or g_{n} with n ≥ 2, (55) is obvious if a_{0} > |a_{1} − a_{2}|; so we may assume that a_{0} = a_{1} − a_{2} > a_{2} − a_{1} (the other case, a_{0} = a_{2} − a_{1}, has a similar proof). Now a_{0} = a_{1} − a_{2} implies that a_{2}γ = 2, which implies that u_{m} = u_{m+1} and v_{m−1} = v_{m} for every odd m > 0. Since n is minimal such that (54) holds, we have τ′ = f_{n} if n is odd and τ′ = g_{n} if n is even. So τ′ ∈ {g_{2}, f_{3}, g_{4}, f_{5}, … } = W^{(a_{0},a_{1},a_{2})} and (55) holds. □

Special pairs

In the following, 𝔸_{∗}^{1} denotes the affine line minus one point. 8.3. Let X be a surface satisfying (‡) and let Λ be an affine ruling of X.

(1) An ordered pair (F, G) of members of Λ (F, G ∈ Λ) is called a special pair of Λ

if (i) F ≠ G, (ii) F ∈ Λ_{∗} and (iii) {F, G} contains all multiple members of Λ. Note the following facts (3 and 4 follow from 1.11 of [2]):

(2) Λ admits a special pair: Λ_{∗} is nonempty and, given F ∈ Λ_{∗}, the definition of Λ_{∗} guarantees that there exists G ∈ Λ such that (F, G) is a special pair.

(3) If (F, G) is a special pair of Λ then X \ supp(F + G) is isomorphic to 𝔸^{1} × 𝔸_{∗}^{1}, in such a way that the projection 𝔸^{1} × 𝔸_{∗}^{1} → 𝔸_{∗}^{1} extends to a rational map X → ℙ^{1} which is compatible with the linear system Λ (i.e., the fibres of the map are members of Λ).

(4) Suppose that U is an open subset of X isomorphic to the product of 𝔸^{1} with some open subset of ℙ^{1}, in such a way that the so obtained rational map X → ℙ^{1} is compatible with Λ. If X \ U contains at least two curves, then there exists a special pair (F, G) of Λ and members M_{1}, …, M_{n} (n ≥ 0) of Λ such that U = X \ supp(F + G + M_{1} + ⋯ M_{n}).

Given a tableau T = ( \begin{matrix}p_{1} & p_{2} & ... & c_{k} \\ c_{1} & c_{2} & ... & c_{k} \end{matrix}) ∈ T , we define (as in 5.35 of [2]) μ(T) = c_{1} ⋯ c_{k} (where μ(T) = 1 if T is the empty tableau). The following is a special case of Corollary 5.37 of [2]: 8.4. Let X be a surface satisfying (‡), let Λ be an affine ruling of X and let (F, G) be a special pair of Λ. If (m, T_{1}, T_{2}) is the discrete part of (X, Λ, F), then

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

where C_{1}, C_{2} ⊂ X are irreducible curves. Moreover, Pic(X_{s}) ≅ ℤ ⊕ ℤ/dℤ, where d = gcd(μ(T_{1}), μ(T_{2})).

Part (1) of the following result was also obtained in [1]: Corollary 8.5. Let the notation be as in 8.4 and suppose that X = ℙ(a_{0}, a_{1}, a_{2}) where a_{0}, a_{1}, a_{2} are pairwise relatively prime. Then

(1) gcd(deg(C_{1}), deg(C_{2})) = 1;

(2) μ(T_{1}) = deg(C_{2}) and μ(T_{2}) = deg(C_{1}).

Proof. We have μ(T_{2}) deg(C_{2}) = μ(T_{1}) deg(C_{1}) and gcd(μ(T_{1}), μ(T_{2})) = 1 by 8.4, so assertions (1) and (2) are equivalent. By part (2) of 2.25 together with the results of sections 5 and 6, there exists a sequence (τ_{0}, …, τ_{n}) in 𝕋(‡) satisfying:

(a) τ_{n} = (m, T_{1}, T_{2}) is the discrete part of (X, Λ, F);

(b) τ_{0} ∈ 𝕋_{I}(a, b, c), for some permutation a, b, c of a_{0}, a_{1}, a_{2};

(c) for each i such that 1 ≤ i ≤ n, the pair (τ_{i−1}, τ_{i}) satisfies one of the following conditions:

(i) τ_{i} > τ_{i−1},

(ii) τ_{i} ∈ E is minimal in 𝕋(‡) and τ_{i−1} = τ̃_{i},
(iii) τ_{i} ∈ E and τ_{i−1} = τ_{i}^{×},
(iv) τ_{i} ∈ 𝕋_{II.2}(a, b, c) and τ_{i−1} = τ_{i}^{×} (some permutation a, b, c of a_{0}, a_{1}, a_{2}).

We proceed by induction on n. If n = 0 then Λ is basic of type I, so C_{1} = R_{i} and C_{2} = R_{j} for some distinct i, j ∈ {0, 1, 2} (notations as in sections 1 and 3). Since gcd(a_{i}, a_{j}) = 1, (1) is clear in this case. Suppose that n > 0 and that (1) (or equivalently (2)) holds for smaller values of n.

If (τ_{n−1}, τ_{n}) satisfies (iii) or (iv) then Λ is basic, so (G, F) is also a special pair of Λ and τ_{n−1} = disc(X, Λ, G); by the inductive hypothesis, (1) holds for Λ and (G, F); it follows immediately that (1) holds for Λ and (F, G).

If (τ_{n−1}, τ_{n}) satisfies (i) or (ii) then τ ≡ τ_{n−1} (by 2.20 or 6.2), so, by 2.25, there exists an affine ruling Λ′ of X and F′ ∈ Λ′_{∗} such that supp(F) = supp(F′) and τ_{n−1} = disc(X, Λ′, F′). Let G′ be such that (F′, G′) is a special pair of Λ′, write τ_{n−1} = (m′, T′_{1}, T′_{2}) and note that 8.4 gives

F′ = μ(T′_{2})C′_{2} and G′ = μ(T′_{1})C′_{1},

where C′_{1} and C′_{2} are irreducible curves. Then

deg(C_{2}) = deg(C′_{2}) = μ(T′_{1}) = μ(T_{1}),

where the middle equality is the inductive hypothesis (i.e., (2) holds for Λ′ and (F′, G′)), the first equality is C_{2} = supp(F) = supp(F′) = C′_{2} and the last equality follows from T′_{1} = T_{1}^{(ˇ s)} for some s ≥ 1. Consequently, (1) and (2) hold for Λ and (F, G). □

Remark. Given ℙ = ℙ(a_{0}, a_{1}, a_{2}), where a_{0}, a_{1}, a_{2} are pairwise relatively prime positive integers, one may ask: What are all pairs of irreducible curves C_{1}, C_{2} ⊂ ℙ with the property that ℙ \ (C_{1} ∪ C_{2}) is isomorphic to the product of 𝔸^{1} with a curve? As mentioned in 8.3, above, these are exactly the special pairs associated to affine rulings of ℙ; consequently, a description of these curves can be derived from this paper. In particular, one can give all pairs of integers (deg(C_{1}), deg(C_{2})) by following these steps:

(1) Give all elements of 𝕋_{0}(ℙ) (7.8 and 7.9 are two examples of this);

(2) for each (m′, T′_{1}, T′_{2}) ∈ 𝕋_{0}(ℙ), give all elements of

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

(this step is computed explicitely in 5.40 of [2]). By 8.5, this set of pairs is the desired one.

For instance, if X = ℙ^{2} = ℙ(1, 1, 1) then one finds that the set of pairs (deg(C_{1}), deg(C_{2})) is the union of the following four sets (where the sequences {u_{n}}_{n=0}^{∞} and {ξ_{n}}_{n=0}^{∞} are defined in 7.8):

(1) (1, n), with n ≥ 1;

(2) (2, 4n + 1), with n ≥ 0; (3) (u_{n}, u_{n+1}P), where n ≥ 3 and (for n fixed) P is any finite product of the form

P = ∏_{i=1}^{s}(α_{i} + u_{n}^{2}ν_{i}) where s ≥ 0, ν_{i} ≥ 0 and α_{i} = \begin{cases}u_{n}(u_{n}−ξ_{n})−1, & if i is odd, \\ u_{n}ξ_{n}−1, & if i is even; \end{cases}

(4) (u_{n+1}, u_{n}Q), where n ≥ 2 and (for n fixed) Q is any finite product of the form

Q = ∏_{i=1}^{s}(α_{i}+u_{n+1}^{2}ν_{i}) where s ≥ 0, ν_{i} ≥ 0 and α_{i} = \begin{cases}u_{n+1}ξ_{n+1}−1, & if i is odd, \\ u_{n+1}(u_{n+1}−ξ_{n+1})−1, & if i is even. \end{cases}

References

1. D. Daigle, On kernels of homogeneous locally nilpotent derivations of k[x, y, z], To appear in Osaka J. Math.

2. D. Daigle and P. Russell, Affine rulings of normal rational surfaces, To appear.

3. 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.

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

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

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

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

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