P-Resolutions of Cyclic Quotients from the Toric Viewpoint

Klaus Altmann

Institut für reine Mathematik, Humboldt-Universität zu Berlin Ziegelstraße 13A, D-10099 Berlin, Germany. E-mail: altmann@mathematik.hu-berlin.de

1 Introduction

(1.1) The break through in deformation theory of (two-dimensional) quotient singularities Y was Kollár/Shepherd-Barron’s discovery of the one-to-one correspondence between so-called Presolutions, on the one hand, and components of the versal base space, on the other (cf. [KS], Theorem (3.9)). It generalizes the fact that all deformations admitting a simultaneous (RDP-) resolution form one single component, the Artin component.

According to definition (3.8) in [KS], P-resolutions are partial resolutions π : Ỹ → Y such that

• the canonical divisor K_{Ỹ |Y} is ample relative to π (a minimality condition) and

• Ỹ contains only mild singularities of a certain type (so-called T-singularities).

Despite their definition as those quotient singularities admitting a I Q-Gorenstein one-parameter smoothing ([KS], (3.7)), there are at least three further descriptions of the class of T-singularities: An explicit list of their defining group actions on I C^{2} ([KS], (3.10)), an inductive procedure to construct their resolution graphs ([KS], (3.11)), and a characterization using toric language ([Al], (7.3)). The latter one begins with the observation that affine, two-dimensional toric varieties (given by some rational, polyhedral cone σ ⊆ IR^{2}) provide exactly the two-dimensional cyclic quotient singularities. Then, T-singularities come from cones over rational intervals of integer length placed in height one (i.e. contained in the affine line (•, 1) ⊆ IR^{2}).

[figure: length 2; σ]

If the affine interval is of length μ + 1, then the corresponding T-singularity will have Milnor number μ (on the I Q-Gorenstein one-parameter smoothing).

(1.2) In [Ch] and [St] Christophersen and Stevens gave a combinatorial description of all P-resolutions for two-dimensional, cyclic quotient singularities. Using an inductive construction method (going through different cyclic quotients with step-by-step increasing multiplicity), they have shown that there is a one-to-one correspondence between P-resolutions, on the one hand, and certain integer tuples (k_{2},…, k_{e−1}) yielding zero if expanded as a (negative) continued fraction (cf. (4.2)), on the other hand.

The aim of the present paper is to provide an elementary, direct method for constructing the Presolutions of a cyclic quotient singularity (i.e. a two-dimensional toric variety) Y_{σ}. Given a chain (k_{2},…, k_{e−1}) representing zero, we will give a straight description of the corresponding polyhedral subdivision of σ. (In particular, the bijection between those 0-chains and P-resolutions will be proved again by a different method.)

2 Cyclic Quotient Singularities

In the following we remind the reader of basic notions concerning continued fractions and cyclic quotients as well as fix notation. References are [Od] (§1.6) or the first sections in [Ch] and [St], respectively.

(2.1) Definition: To integers c_{1},…, c_{r} ∈ ZZ we will assign the continued fraction [c_{1},…, c_{r}] ∈ I Q if the following inductive procedure is well-defined (i.e. if no division by 0 occurs):

• [c_{r}] := c_{r}

• [c_{i},…, c_{r}] := c_{i} − 1/[c_{i+1},…, c_{r}].

If c_{i} ≥ 2 for i = 1,…, r, then [c_{1},…, c_{r}] is always defined and yields a rational number greater than 1. Moreover, all these numbers may be represented by those continued fractions in a unique way.

(2.2) Let n ≥ 2 be an integer and q ∈ (ZZ/nZZ)^{∗} be represented by an integer between 0 and

n. Each q provides a group action of ZZ/nZZ on I C^{2} via the matrix ( \begin{matrix}ξ & 0 \\ 0 & ξ^{q} \end{matrix} ) (with ξ a primitive

n-th root of unity). The quotient is denoted by Y (n, q). In toric language, Y (n, q) equals the variety Y_{σ} assigned to the polyhedral cone σ := ⟨(1, 0); (−q, n)⟩ contained in IR^{2}. Y_{σ} is defined as Spec I C[σ^{∨} ∩ ZZ^{2}] with

σ^{∨} := {r ∈ (IR^{2})^{∗} | r ≥ 0 on σ} = ⟨[0, 1]; [n, q]⟩ ⊆ (IR^{2})^{∗} ≅ IR^{2} .

Notation: Just to distinguish between IR^{2} and its dual (IR^{2})^{∗} ≅ IR^{2}, we will denote these vector spaces by N_{IR} and M_{IR}, respectively. (Hence, σ ⊆ N_{IR} and σ^{∨} ⊆ M_{IR}.) Elements of N_{IR} ≅ IR^{2} are written in parentheses; elements of M_{IR} ≅ IR^{2} are written in brackets. The natural pairing between N_{IR} and M_{IR} is denoted by ⟨ , ⟩, which should not be confused with the symbol indicating the generators of a cone. Finally, all these remarks apply for the lattices N ≅ ZZ^{2} and M ≅ ZZ^{2}, too.

(2.3) Let n, q be as before. We may write n/(n − q) and n/q (both are greater than 1) as continued fractions

n/(n − q) = [a_{2},…, a_{e−1}] and n/q = [b_{1},…, b_{r}] (a_{i}, b_{j} ≥ 2).

The a_{i}’s and the b_{j}’s are mutually related by Riemenschneider’s point diagram (cf. [Ri]). Denote by w^{1}, w^{2},…, w^{e} the lattice points on the compact edges of the convex hull of (σ^{∨}∩M)\{0}. If ordered the right way, we obtain w^{1} = [0, 1] and w^{e} = [n, q] for the first and the last point, respectively.

[figure: 0; w^{1}; w^{2}; …; w^{3}; w^{e}; σ^{∨}]

Then, E := {w^{1},…, w^{e}} is the minimal generating set (the so-called Hilbert basis) of the semigroup σ^{∨} ∩ M. These point are related to our first continued fraction by

w^{i−1} + w^{i+1} = a_{i} w^{i} (i = 2,…, e − 1).

Remark: The surjection IN^{E} ⟶ ⟶ σ^{∨} ∩ M provides a minimal embedding of Y_{σ}. In particular, e equals its embedding dimension.

In a similar manner we can define v^{0},…, v^{r+1} ∈ σ∩N in the original cone; now we have v^{0} = (1, 0), v^{r+1} = (−q, n), and the relation to the continued fractions is v^{j−1} + v^{j+1} = b_{j} v^{j} (for j = 1,…, r).

[figure: v^{0}; v^{1}; v^{r+1}; 0; σ]

Drawing rays through the origin and each point v^{j}, respectively, provides a polyhedral subdivision Σ of σ. The corresponding toric variety Y_{Σ} is a resolution of our singularity Y_{σ}. The numbers −b_{j} equal the self intersection numbers of the exceptional divisors; since b_{j} ≥ 2, the resolution is minimal.

3 The Maximal Resolution

(3.1) Definition: ([KS], (3.12)) For a resolution π : Ỹ → Y we may write K_{Ỹ |Y} := KỸ − π^{∗}K_{Y} = ∑_{j}(α_{j} − 1)E_{j}, where the E_{j} denote the exceptional divisors, and α_{j} ∈ I Q. Then, π will be called maximal if it is maximal with respect to the property 0 < α_{j} < 1.

The maximal resolution is uniquely determined and dominates all the P-resolutions. Hence, for our purpose, it is more important than the minimal one. It can be constructed from the minimal resolution by successively blowing up points E_{i} ∩ E_{j} with α_{i} + α_{j} ≥ 0 (cf. Lemma (3.13) and Lemma (3.14) in [KS]).

(3.2) Proposition: The maximal resolution of Y_{σ} is toric. It can be obtained by drawing rays through 0 and all interior lattice points (i.e. ∈ N) of the triangle Δ := conv (0, v^{0}, v^{r+1}), respectively.

Proof: In order to keep track of the rational numbers α_{j}, we will show how they can be “seen” in an arbitrary toric resolution of Y_{σ}. Let Σ < σ be a subdivision generated by one-dimensional rays through the points u^{0},…, u^{s+1} ∈ σ∩N. In particular, u^{0} = v^{0} = (1, 0) and u^{s+1} = v^{r+1} = (−q, n); moreover, for the minimal resolution we have s = r and u^{j} = v^{j} (j = 0,…, r + 1). Denote by c_{1},…, c_{s} the integers given by the relations

u^{j−1} + u^{j+1} = c_{j} u^{j} (j = 1,…, s).

In particular, c_{j} = b_{j} for the minimal resolution. As usual, the numbers −c_{j} equal the self intersection numbers of the exceptional divisors E_{j} in Y_{Σ}. Indeed, D := ∑_{i} u^{i} E_{i} is a principal divisor (if you do not like coefficients u^{i} from N, evaluate them using arbitrary elements of M); hence,

0 = E_{j} · D = E_{j} · (u^{j−1}E_{j−1} + u^{j}E_{j} + u^{j+1}E_{j+1})
= u^{j−1} + (E_{j})^{2} u^{j} + u^{j+1}
= (c_{j} + (E_{j})^{2}) · u^{j}  (j = 1,…, s).

On the other hand, we can use the projection formula to obtain

−2 = 2 g(E_{j}) − 2 = K_{Ỹ |Y} · E_{j} + (E_{j})^{2}
= ∑_{i} (α_{i} − 1) (E_{i} · E_{j}) + (E_{j})^{2}
= (α_{j−1} − 1) + (α_{j} − 1) (E_{j})^{2} + (α_{j+1} − 1) + (E_{j})^{2} ,

hence

α_{j−1} + α_{j+1} = c_{j} α_{j} (j = 1,…, s; α_{0}, α_{s+1} := 1).

Looking at the definition of the c_{j}’s via relations among the lattice points u^{j}, there has to be some R ∈ M_{IR} such that

α_{j} = ⟨u^{j}, R⟩ (j = 0,…, s + 1).

The conditions ⟨u^{0}, R⟩ = α_{0} = 1 and ⟨u^{s+1}, R⟩ = α_{s+1} = 1 determine R uniquely. Now, we can see that α_{j} measures exactly the quotient between the length of the line segment \overline{0 u^{j}} and the length of the Δ-part of the line through 0 and u^{j}. In particular, α_{j} < 1 if and only if u^{j} lies below the line connecting u^{0} and u^{s+1}.

[figure: 0; u^{0}; u^{i} with α_{i} > 1; α_{j} < 1; u^{s+1}; line [R = 1]; σ]

This explains how to construct the maximal resolution: Starting with the minimal one, continue subdividing each small cone ⟨u^{j}, u^{j+1}⟩ into ⟨u^{j}, u^{j} +u^{j+1}⟩ ∪ ⟨u^{j} +u^{j+1}, u^{j+1}⟩ as long as it contains interior lattice points below the line [R = 1], i.e. belonging to int Δ. □

Corollary: Every P-resolution is toric.

Proof: P-resolutions are obtained by blowing down curves in the maximal resolution. □

(3.3) Example: We take the example Y (19, 7) from [KS], (3.15). Since σ = ⟨(1, 0), (−7, 19)⟩, the interior of Δ is given by the three inequalities

y > 0 , 19x + 7y > 0 , and 19x + 8y < 19 (corresponding to R = [1, 8/19]) .

The only primitive lattice points (i.e. generating rays) contained in int Δ are

u^{1} = (0, 1) , u^{2} = (−1, 4) , u^{3} = (−2, 7) , u^{4} = (−1, 3) , u^{5} = (−5, 14) , u^{6} = (−4, 11) .

They provide the maximal resolution. The corresponding α’s can be obtained by taking the scalar product with R = [1, 8/19], which yields 8/19, 13/19, 18/19, 5/19, 17/19, and 12/19. The minimal resolution uses only the rays through u^{1} = (0, 1), u^{4} = (−1, 3), and u^{6} = (−4, 11), respectively.

4 P-Resolutions

(4.1) In this section we will speak about partial toric resolutions π : Y_{Σ} → Y_{σ}. Nevertheless, we use the same notation as we did for the maximal resolution: The fan Σ subdividing σ is generated by rays through u^{0},…, u^{s} ∈ σ ∩ N; each ray u^{j} corresponds to an exceptional divisor E_{j} ⊆ Y_{Σ}. However, since u^{j−1} + u^{j+1} need not to be a multiple of u^{j}, the numbers c_{j} no longer make sense.

Lemma: ([Re], (4.3)) For K := K_{Y_{Σ}} or K := K_{Y_{Σ}|Y_{σ}} the intersection number (E_{j} · K) is positive, zero, or negative if the line segments \overline{u^{j−1}u^{j}} and \overline{u^{j}u^{j+1}} form a strict concave, flat, or strict convex “roof” over the two cones, respectively.

[figure: u^{j+1}; u^{j−1}; (E_{j} · K) > 0; u^{j+1}; u^{j−1}; (E_{j} · K) = 0; u^{j+1}; u^{j−1}; 0; (E_{j} · K) < 0; u^{j}; u^{j}; u^{j}; 0; 0]

Proof: Using K := K_{Y_{Σ}} = − ∑_{i=0}^{s+1} E_{i} (cf. [Od], (2.1)) we have

(E_{j} · K) = −(E_{j} · E_{j−1}) − (E_{j})^{2} − (E_{j} · E_{j+1}) .

On the other hand, as in the proof of Proposition (3.2), we know that

0 = (E_{j} · E_{j−1}) u^{j−1} + (E_{j})^{2} u^{j} + (E_{j} · E_{j+1}) u^{j+1} .

Combining both formulas yields the final result

(E_{j} · K) u^{j} = (E_{j} · E_{j−1}) (u^{j−1} − u^{j}) + (E_{j} · E_{j+1}) (u^{j+1} − u^{j}) . □

Remark: The previous lemma together with Proposition (3.2) illustrate again the fact that all P-resolutions are dominated by the maximal resolution.

(4.2) In [Ch] Christophersen has defined the set

K_{e−2} := {(k_{2},…, k_{e−1}) ∈ IN^{e−2} | [k_{2},…, k_{e−1}] is well defined and yields 0 }

of chains representing zero. To every such chain, non-negative integers q_{1},…, q_{e} are assigned. They are characterized by the following equivalent properties:

• q_{1} = 0, q_{2} = 1, and q_{i−1} + q_{i+1} = k_{i} q_{i} (i = 2,…, e − 1);

• q_{e−1} = 1, q_{e} = 0, and q_{i−1} + q_{i+1} = k_{i} q_{i} (i = 2,…, e − 1);

• q_{e} = 0 and [k_{i},…, k_{e−1}] = q_{i−1}/q_{i} with gcd(q_{i−1}, q_{i}) = 1 (i = 2,…, e − 1).

(The two latter properties do not even use the fact that the continued fraction [k_{2},…, k_{e−1}] yields zero.)

Remark: The elements of K_{e−2} are in one-to-one correspondence with triangulations of a (regular) (e−1)-gon with vertices P_{2},…, P_{e−1}, P_{∗}. The numbers k_{i} tell how many triangles are attached to P_{i}, and the q_{i} have an easy meaning in this language, too.

Finally, for a given Y_{σ} with embedding dimension e, Christophersen defines

K(Y_{σ}) := {(k_{2},…, k_{e−1}) ∈ K_{e−2} | k_{i} ≤ a_{i}} .

Theorem: Each P-resolution of Y_{σ} (i.e. the corresponding subdivision Σ of σ) is given by some \underline{k} ∈ K(Y_{σ}) in the following way:

(1) Σ is built from the rays that are orthogonal to w^{i}/q_{i} − w^{i−1}/q_{i−1} ∈ M_{IR} (for i = 3,…, e − 1). In some sense, if the occurring divisions by zero are interpreted properly, Σ may be seen as dual to the Newton boundary generated by w^{i}/q_{i} ∈ σ^{∨} (i = 1,…, e).

(2) The affine lines [⟨•, w^{i}⟩ = q_{i}] form the “roofs” of the Σ-cones. In particular, the (possibly degenerate) cones τ^{i} ∈ Σ correspond to the elements w^{1},…, w^{e} ∈ E. The “roof” over the cone τ^{i} has length ℓ_{i} := (a_{i} − k_{i}) q_{i}, where the length is defined via the metric induced by the lattice structure M ⊆ M_{IR} on rational lines. In particular, τ^{i} is degenerate if and only if k_{i} = a_{i}. The Milnor number of the T-singularity Y_{τ^{i}} equals (a_{i} − k_{i} − 1).

(4.3) Proof: According to the notation introduced in (4.1), the fan Σ consists of (nondegenerate) cones τ^{j} := ⟨u^{j−1}, u^{j}⟩ with j = 1,…, s + 1. (Except for u^{0} = (1, 0) and u^{s} = (−q, n), their generators u^{j} are primitive lattice points (i.e. ∈ N) contained in intΔ ⊆ σ.)

Step 1: For each τ^{j} there are w ∈ E, d ∈ IN such that ⟨u^{j−1}, w⟩ = ⟨u^{j}, w⟩ = d. First, it is very clear that there is a primitive lattice point w ∈ M and a non-negative number d ∈ IR_{≥0} admitting the desired properties. Moreover, since u^{j} ∈ N, d has to be an integer, and Reid’s Lemma (4.1) tells us that w ∈ σ^{∨}. It remains to show that w belongs to the Hilbert basis E ⊆ σ^{∨} ∩ M as well. Denote by ℓ the length of the line segment \overline{u^{j−1}u^{j}} on the “roof” line [⟨•, w⟩ = d]. Since τ^{j} represents a T-singularity, we know from (7.3) of [Al] (cf. (1.1) of the present paper) that d|ℓ. In particular, \overline{u^{j−1}u^{j}} contains the d-th multiple d · u of some lattice point u ∈ τ^{j} ∩ M (w.l.o.g. not belonging to the boundary of σ). Hence, ⟨u, w⟩ = 1 and u ∈ int σ ∩ M, and this implies w ∈ E.

Step 2: As each of the cones τ^{1},…, τ^{s+1} ∈ Σ is assigned to some element w ∈ E, a renumbering will be used to make the notation more obvious: Let τ^{i} = ⟨u^{i−1}, u^{i}⟩ be the cone assigned to w^{i} ∈ E, and denote by d_{i}, ℓ_{i} the height and the length of its “roof” \overline{u^{i−1}u^{i}}, respectively. Some of these cones might be degenerated, i.e. ℓ_{i} = 0. This it at least true for the extremal τ^{1} and τ^{e} coinciding with the two rays spanning σ. Here we even have d_{1} = d_{e} = 0; in particular u^{0} = u^{1} = (1, 0) and u^{e−1} = u^{e} = (−q, n). Since d_{i}|ℓ_{i}, we may introduce integers k_{i} ≤ a_{i} yielding ℓ_{i} = (a_{i} − k_{i}) d_{i}. For i = 2,…, e − 1 they are even uniquely determined.

Step 3: Using the following three ingredients

(i) ⟨u^{i−1}, w^{i}⟩ = ⟨u^{i}, w^{i}⟩ = d_{i} (i = 1,…, e) ,

(ii) w^{i−1} + w^{i+1} = a_{i} w^{i} (i = 2,…, e − 1; cf. (2.3)), and

(iii) ⟨u^{i} − u^{i−1}, w^{i−1}⟩ = ℓ_{i} = (a_{i} − k_{i}) d_{i} (since {w^{i−1}, w^{i}} forms a ZZ-basis of M),

we obtain

d_{i−1} + d_{i+1} = (a_{i} d_{i} + d_{i−1}) − (a_{i} d_{i} − d_{i+1})
= (a_{i} d_{i} + ⟨u^{i−1}, w^{i−1}⟩) − ⟨u^{i}, a_{i} w^{i} − w^{i+1}⟩
= a_{i} d_{i} + ⟨u^{i−1}, w^{i−1}⟩ − ⟨u^{i}, w^{i−1}⟩
= a_{i} d_{i} + ⟨u^{i−1} − u^{i}, w^{i−1}⟩
= a_{i} d_{i} − (a_{i} − k_{i}) d_{i} = k_{i} d_{i} (for i = 2,…, e − 1) .

In particular, k_{i} ≥ 0 (and in fact ≥ 1 for e > 3). Moreover, because {w^{i−1}, w^{i}} forms a basis of M and u^{i−1} ∈ N is primitive, we have gcd(d_{i−1}, d_{i}) = 1. It follows that d_{i} = q_{i}, since both sequences of integers satisfy the second of the three properties mentioned in the beginning of (4.2). Finally, the third of these properties yields [k_{2},…, k_{e−1}] = q_{1}/q_{2} = d_{1}/d_{2} = 0, i.e. \underline{k} ∈ K_{e−2}.

The reverse direction, i.e. the fact that each K(Y_{σ})-element indeed yields a P-resolution, follows from the above calculations in a similar manner. □

Remark: Subdividing each τ^{i} further into (a_{i} − k_{i}) equal cones (with each “roof” length q_{i}) yields the so-called M-resolution (cf. [BC]) assigned to a P-resolution. It is defined to contain only T_{0}-singularities, i.e. T-singularities with Milnor number 0; in exchange, K_{Ỹ |Y} does not need to be relatively ample anymore. This property is replaced by “relatively nef”.

Examples: (1) The continued fraction [1, 2, 2,…, 2, 1] = 0 yields q_{1} = q_{e} = 0 and q_{i} = 1 otherwise. In particular, the “roof” lines equal [⟨•, w^{i}⟩ = 1] (for i = 2,…, e − 1), describing the RDP-resolution of Y_{σ}. The assigned M-resolution equals the minimal resolution mentioned at the end of (2.3).

(2) Let us return to Example (3.3): The embedding dimension e of Y_{σ} is 6, the vector (a_{2},…, a_{e−1}) equals (2, 3, 2, 3), and, except the trivial RDP element mentioned in (1), K(Y_{σ}) contains only (1, 3, 1, 2) and (2, 2, 1, 3). In both cases we already know that q_{1} = q_{6} = 0 and q_{2} = q_{5} = 1. The remaining values are given by the equation q_{3}/q_{4} = [k_{4}, k_{5}], i.e. we obtain q_{3} = 1, q_{4} = 2 or q_{3} = 2, q_{4} = 3, respectively. Hence, in case of (1, 3, 1, 2) the fan Σ is given by the additional rays through (0, 1) and (−4, 11). For \underline{k} = (2, 2, 1, 3) we only need the one through (−1, 4).

References

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