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DIMITRIOS I . DAIS AND MARTIN HENK

On a series of Gorenstein cyclic quotient singularities admitting a unique

projective crepant resolution

Preprint SC 97-39 (September 1997)

On a series of Gorenstein yclic q u o t n t singularities admitting a unique

rojective crepant resolution

Dimitrios I. Dais and Martin Henk

ABSTRACT. Let G be a finite subgroup of SL(r, C). In dimensions r = 2 and r = 3, McKay correspondence provides a natural bijection between the set of ir reducible representations of G and a cohomology-ring basis of the overlying space of a projective, crepant desingularization of C jG. For r = 2 this desingulariza-tion is unique and is known to be determined by the Hilbert scheme of the G-orbits. Similar statements (including a method of distinguishing just one among all possible smooth minimal models of C3 /G), are very probably true for all G' C SL(3, C) too, and recent Hilbert-scheme-techniques due to Ito, Nakamura and Reid, are expected to lead to a new fascinating uniform theory. For dimensions r > 4, however, to apply analogous techniques one needs extra modifications. In addition, minimal models of C jG are smooth only under special circumstances C4 / (involution), for instance, cannot have any smooth minimal model. On the other hand, all abelian quotient spaces which are c.i.'s can always be fully resolved by torus-equivariant, crepant, projective morphisms. Hence, from the very begin­ning, the question whether a given Gorenstein quotient space C /G, r > 4 admits special desingularizations of this kind, seems to be absolutely crucial. In the present paper, after a brief introduction to the existence-problem of such desingularizations (for abelian G's) from the point of view of toric geometry we prove that the Gorenstein cyclic quotient singularities of type

y ( , . . . , ( f ) )

with / > r > 2, have a unique torus-equivariant projective, crepant partial res olution, which is "full" iff either / = 0 mod (r — 1) or i = 1 mod (r — 1). As it turns out, if one of these two conditions is fulfilled, then the exceptional locus of the full desingularization consists of [—zj\ prime divisors, [-~zj\ — 1 of which are isomorphic to the total spaces of F^-bundles over IP^-2. Moreover, it is shown that intersection numbers are computable explicitly and that the resolution morphism can be viewed as a composite of successive (normalized) blow-ups. Obviously the monoparametrized singularity-series of the above type contains (as its "first member") the well-known Gorenstein singularity defined by the origin of the affine cone which lies over the r tup le Veronese embedding of P ^ - 1

(a) Let / : Y —> X be a birational morphism between two normal, Q-Gorenstein complex varieties X and Y of index j . Denote by LUX = O {Kx) and toy = O (Ky) the dualizing sheaves, and by Kx and Ky representatives of canonical divisors of X

and Y, respectively. / is called crepant if wx' = /* ( OJ^3 J, or, in other words, if the

discrepancy jKY — f* (jKx) vanishes. The "prototype" for a crepant morphism is the proper birational map which desingularizes the usual doublepointlocus

= {(zzz) e C 3 \( + zl+=Q}

by blowing u p O e l c C 3 . Crepant birational morphisms were mainly used in the past two decades in algebraic geometry to reduce the singularities of com­plex 3-folds (and, sometimes, n-folds) to terminal (or even Q-factorial terminal) singularities, and to treat of minimal models in high dimensions. For X being the underlying space of a Gorenstein quotient singularity they are "by definition" related to McKay-type correspondences

(b) Let r be an integer > 2, G a finite subgroup of GL(r, C) containing no pseu-doreflections and acting linearly on C r , and p : C —> C /G the corresponding quotient map. The underlying space Cr )G of the (germ of the) quotient singular ity ( C / G , [0]), with [0] := p(0), is canonically equipped with the structure of a normal, Cohen-Macaulay complex variety (or complex-analytic space).

• The singular locus Sing(Cr/G) of C / G itself contains always [0], but for r > 3 it is possible to possess also other strata of C of codimension > 2 passing through 0] (cf. 5.1 below).

As it was proved by Watanabe [78], C / G is Gorenstein iff G C SL(r, C).

• If r = 2, G c GL(2C), the quotient space C2 /G admits a unique minimal desingularization

f:X^X = C/G (1

("minimal" in the sence that the exceptional locus of / does not contain any curve with self-intersection number — 1, or equivalently, that there exists, up to isomor phism, a unique morphism h : X —> X with g = f o h, for any desingularization g : X —>• X of X). The description of the prime divisors (rational curves) con­sisting the exceptional locus of the above / , as well as that of the way of how these divisors intersect each other (tree configurations), is due to Hirzebruch [31] (for cyclic acting groups) and Brieskorn [9] (for all the other finite subgroups G of GL(2,C)).

• The minimal desingularization (1.1) is crepant if and only if G C SL(2,C). In this special case, the (Gorenstein) quotient spaces C2 jG are embeddable as A-D-E hypersurfaces in C3 (Klein [41], Du Val [15], [16]) and are nothing but the rational double points treated in the classical theory of the simple hypersurface singulari­ties. (For details, see e.g. Lamotke [45] and Slodowy [71]). In the late seventies McKay [48] observed a remarkable connection between the representation theory

of the finite subgroups of SL(2, C) and the Dynkin diagrams of certain irreducible root systems. Having this as starting-point, Gonzalez-Sprinberg, Verdier [24] and later Knörrer [43], constructed a purely geometric, direct correspondence "of McKay-type" between the set of irreducible representations of G, and the coho mology ring of X. Recently, Ito, Nakamura [38], and Reid [62], introduced new techniques for the study of McKay correspondence involving tautological sheaves and Hilbert schemes of G-orbits.

In particular, for r = 2, the main result of Ito and Nakamura [ ] [37] [38] [ ] [5] can be roughly stated as follows:

Theorem 1.1. Let G be a unite subgroup of SL(2,C) with = \G\. Then there is a unique irreducible component T~LG (C2) of the G-fixed point set V} [C2 ] of the Hilbert scheme H [€?] parametrizin ll uste of length C2 such that the induced proper brational morphis

/G = X

gives again the minima resolution (1.1) of X (up to isohism). Moreover, the original correspondence of [48], [24], [43], between the nn-trivial representations of G and the exceptiona prme divisor of ( can eintereted excusivel in terms of suitable idea of %G C2)

• More generally, for arbitrary r, if / : X —> X denotes a projective crepant ("full") desingularization of X = C / G , then the expected bijections are those of the following box

irreducible representations of the group G

t conjugacy 1 classes of G J

a suitable basis of the cohomology

r ing i (X;Z

a suitable basis of the

homology ring H. (X; Z)

ic is ow k id logan :

representation theory of G " = " homology of X.

This conjecture is accompanied by the remark that the above bijections probably satisfy certain "compatibilities" (as for r = 2) with respect to the behaviour of the cup product, the image of the character table of G, the duality interrelation etc.

• Although on the level of "counting" dimensions of rational cohomology groups or even on that of providing formal correspondences between the left and the right

hand side, many things are more or less well-understood (by to i c methods [4 for G abelian, and by results of Batyrev [2], [3] and Kontsevich [44] involving p-adic integration and loop spaces, for arbitrary G's), there are still lots of open questions of how one might work with (co)homology groups the coefficients of which are taken from Z.

• Reid's approach to this generalized McKay-type-conjecture is two-fold. The first idea (concerning correspondence B) relies on the application of the following Ito-Reid theorem [39] in order to construct a suitable collection of loci within X (ie a collection of centers of monomial valuations on C(X)) generating H* IX; Z J.

Theorem 1.2. LetG be a unite subgroup of SL(r C) acting linearly on C*, r > 2 and X = Cr JG. Then thee is a canonical one-to-one correspondence etween the junior conjugac classes in and the crepant discrete valuations of X

• The second idea (w.r.t. bijection A) is to consider the tautological sheaves $p

assigned to the irreducible representations p of G. Reid [62] conjectures (and proves for several examples) that appropriate Z-linear combinations of the Chern classes

of 3p's lead to a canonical Z-basis of the cohomology ring H ( X ; Z ) . Moreover

if X happens to be isomorphic to %G (C*), then these sheaves enjoy very good algebraic-geometric properties (they are generated by their global sections, are vector bundles, their first Chern classes induce nef linear systems etc., cf. [62] 5.5). In particular, in this case HG (Cr) is birationally distinguished among all the other projective crepant resolutions of X.

• As both A and B are clarified in dimension 2, let us recall what is known for r = 3. Next theorem is due to Markushevich ], [47], Ito [33], [34], [35], and Roan [4] , [5] , [66], [67]

Theorem 1.3. The underlying spaces of ll 3 -diensonal Gorenstein qutient sinrities possess repant resolutions

onjecture 1.4. Let G be an finite subgroup of SL(3C). Then UG (C3) is repant resolutio ofCs/G.

For abelian G's Conjecture 1.4 was proved by Nakamura [52] (for an outline of the proof see also [62], § 7); work on the non-abelian case is in progress. The complete verification of 1.4 would mean that for all quotients C 3 /G there is always a dis tinguished1 smooth minimal model available, satisfying all the above mentioned peculiar properties.

• In dimensions r > 4, however, there are certain additional troubles already from the very beginning and Reid's complementary question (61], [39], § 4 5 2] 54) still remains an unanswered enigma:

1In contrast to dimension 2, in dimension 3 minimal models are unique only up to isomor phisms in codimension 1. Moreover, there exists lots of examples of acting groups G, for which C?/G has crepant, full, non-projective resolutions (see below 7.5)

Reid's q e s t i o n : U n d r whic onditions on e acting g o u p s

c SL (r C) , r > 4

do the quotient spaces C jG have projective crepant desingularizations?

• Note that the existence of terminal Gorenstein singularities implies automatically that not all Gorenstein quotient spaces C / G , r > 4 can have such desingulariza tions (cf. Morrison-Stevens [49]).

• Moreover, in contrast to what is valid in the "low" dimensions 2 and 3, the Hilbert scheme 'HG (Cr) for r > 4 might be singular, even if the quotient C jG being under consideration is known to possess projective, crepant resolutions. (We are indebted to I.Nakamura and M.Reid for this information2)

• On the other hand, as it was proved in [12] by making use of Watanabe's clas sification [79] of all abelian quotient singularities (C r /G, [0]), G C SL(r, C), (up to analytic isomorphism) whose underlying spaces are embeddable as complete intersections ("c.i.'s") of hypersurfaces into an affine complex space and methods of toric and discrete geometry

Theorem 1.5. The underlying spaces of all abelian quotient c.-singularities ad mit of torus-equivariant projective epant resolutions and theefor smooth min­imal models) in all dimensons

In particular, taking into account the specific structure of these singularities de pending on the free parameters of the so-called "Watanabe forests", this theorem guarantees the existence of infinitely many (isomorphism classes of) Gorenstein quotient singularities in each dimension having resolutions with the required prop­erties. Nevertheless, these c.i.-singularities are of special nature, and they form a relatively "sparse" subclass of the class of all Gorenstein abelian quotient singular ities (For instance, all Gorenstein cyclic quotient msc-singularities in dimensions > 3 are not c.i.'s!). Thus, as "next step" it is natural to ask what happens with respect to the non-c.i.'s. Various necessary existence-criteria working quite well in the framework of this most general consideration, partially sufficient conditions and certain theoretical and algorithmic difficulties which arise from LP-feasibility problems (cf. rem. 6.16 below), as well as further families of non-c.i. abelian quo tient singularities for which it is possible to apply a direct, constructive method to obtain the desired resolutions, will be discussed in detail in [13]. In the present paper, we shall study another special (but, again, infinite) series of Gorenstein non-c.i. (for r > 3), cyclic quotient singularities admitting a uniquely determine torus-equivariant projective, crepant resolution under very simple and absolutely well-controllable (necessary and sufficient) number-theoretic conditions. This res olution will be defined as an immediate generalization of the most well-known example in the literature namely of the "single blow-up" of the affine cone over

2Nakamura's typical counterexample is the so-called (4; 2)-hypersurface-singularity (in the terminology of [12]), with non-smooth 'HG. This singularity has projective crepant resolutions (cf. [12], cor. 6.3, or [65], § 5).

the /-tuple Veronese embedding of r at the origi et us first formulate it explicitly

Proposition 1.6. Let G be the unite clic group of analytic automorphis of C r > 2 of orde > 2 generated by

g : C 3 ( z z ) i (e^~^~ e^~^~ G C 2)

(i) (Cr /G, [0]) is an isolated sinrity and its underlyin space is embedded int

as the zeroset

{£u tv • • • t ti ti • • • twi | sort ( • • • ) = sort (u' v' • • w') } ,

where u re defined as tie index sets

= U t 11 . 2 2 . 2 3 3 . 3 . rr . r i-times 2-times ^-times -times

with i2\ r = j I, Vj, 1 r,

and sort( denotes the sorting of an strin of the alphaet 2 3 r} int weakly increasing order.

(ii) The sinrity ( C / G , [0]) is

terminal < = r >

canonical <S= r >

Gorenstein <£= \ r

(iii) If Bi ( C ) denotes the (usual) blow-up ofCr at the orgin then the acti of G Cr can e extended ont i ( C )

( C ) / G 7 G 3)

is a (full) resolution of [0] € C jG , the exceptional prim divisor D is isomorphic o P^ - 1 , and the corresponding rlative canonical divisor equa ( j —

(iv) (1.3) is (the unique) epant resolution ofC/G f and if r =

Proof, (i) is an easy exercise (one has just to compute the generators of the ring of invariants and their relations, cf. [40], p. 40, and Sturmfels [72], p. 1, for the explanation); (ii) follows from the general theorems of Reid [58] (cf. (1.5), p 277, (3.1), p. 292) or directly from (iii). The construction of (1.3) is due to Ueno [74] (see also [75] pp. 99211) who called it the "canonical resolution" of C / G

and used it to obtain generalized Kummer manifolds by resolving special quotien spaces denned by discrete groups acting on complex tori, (iv) was pointed out by Hirzebruch & Höfer in [32] p. 257, and follows from (iii). For the uniqueness (up to isomorphism), see e.g. Roan's comments in [67], Ex p. 135. D

(c) For fixed dimension r > 2 a "2-parameter" series of cyclic quotient singulari ties containing ( 2 ) of prop. as its "first member" (ß = ) is that of type

j , . .

\ (r — l)-times

(see 5.3 for the definition and notation). A unique "canonical desingularization" for each of its members was given by Ueno [73], § 4, pp. 53-63, and Fujiki [20] pp. 316-318, in the case in which the dimension equals r = 3, and turned out to be very useful for the characterization of exceptional fibers belonging to smooth threefolds fibered over a curve and having normally polarized abelian surfaces as generic fibers. (This was, in fact, a direct generalization of Hirzebruch's "contin­ued fraction algorithm" [31], pp. 15-20, to the next coming dimension). Since we are mainly interested in Gorenstein singularities and in the existence (or non­existence) of projective, crepant, full resolutions in dimensions r > 4, we shall consider for (1.4) only the case where ß = I — (r — 1) (cf. 5.6, 8.1 below) and con­sequently I as the only available parameter. Our main motivation to work out this series was a recent remark of Reid in [62], 5.4; namely, that the four-dimensional cyclic singularities of type j (1,1,1, / —3) with gcd(£,3) = 1 are to be resolved by a crepant morphism if and only if I = 1 mod 3. We give a generalization in all dimensions, even without assuming the isolatedness of the corresponding sin­gularity, show the uniqueness and projectivity of the crepant morphism, describe the exceptional prime divisors and their intersection numbers, and compute the cohomology dimensions of the desingularized space.

Exactly as in [12], we shall exclusively work with the machinery of toric geometry More precisely, the paper has the following structuring : In section 2 we give the toric glossary which will be used in the sequel. (The reader who is familiar with this matter may skip it). Sections 3 and 4 are complementary. (In fact, the reason for adding in § 3 some lengthy explanations is that there is a potential for confu­sion between the usual blow-up of a toric subvariety V (r) of an X (N, A) and the starring subdivision w.r.t. r . These are identical only for smooth X (N, A)'s! On the other hand, to lend an algebraic-geometric characterization to even very simple combinatorially motivated cone subdivisions, it is absolutely natural to blow-up also not necessarily reduced subschemes). In § 4 we deal with a high dimensional analogue of the so-called Hirzebruch-surfaces and make certain remarks concerning its embeddings and intersection theory. (It turns out that all but one exceptional prime divisors which will arise later on in our desingularizations are of this sort) Sections 5-6 outline a first systematic approach to the general problem of the exis tence or non-existence of crepant (preferably projective) resolutions of Gorenstei

abelian quotient singularities o dimension > 4 by basic (and coherent) triangu-lations of the junior simplex. In § 7 we take a closer look at the low dimensions Sections 8 and 9 contain our main results. Though the singularity-series which we study is rather special, we hope at least that it will become clear how one may apply our techniques to more demanding singularities. In particular, in section 9 the factorization of the desingularizing morphism is reduced to a "game" with the available simplices. Finally, in § 10 we give a foretaste of what may be done for the series generalizing j (1 ,24) and state the GPSS-conjecture.

Terminology and general notation. By a complex variety is meant an integral, separated algebraic scheme over C. A complex variety is therefore an irreducible reduced ringed space (X, Ox) with structure sheaf Ox which is locally determined by the canonical structure sheaf of the spectrum of an affine complex coordinate ring. Sing(X) denotes the singular locus of X, ie. , the set of all points x £ X with öx,x a non-regular local ring. Analogously, i 6 l i s normal, Cohen-Macaulay Gorenstein etc., if Ox,x is of this type. A subvariety Y of X is a closed integral subscheme of X. If codim^ (Y) = 1, then Y is especially called a prime divisor By CDiv(X), WDiv(X), Pic(X) and A, (X) = ®k>oAk (X) we denote the groups of Cartier and Weil divisors, the Picard group, and the graded Chow ring of X, re­spectively. (For X smooth, A* (X) = ®k>0A (X), with Ak (X) = AdimX-k (X)) Just as in [4], [12], by a desingularization (or resolution of singularities) f : X —> X of a non-smooth X, we mean a "full" or "overall" desingularization (if not men­tioned), i.e., Sing(X) = 0 . When we deal with partial desingularizations, we

mention it explicitly. A birational morphism / : X' —l X is projective if X' ad­mits an /-ample Cartier divisor. The intersection numbers of Cartier divisors are defined as in [ ] (see below § 2 ( i )

2. Preliminaries from toric geometry

We recall some basic facts from the theory of toric varieties and fix the notation which will be used in the sequel. For details the reader is referred to the standard textbooks of Oda [53], Fulton [22] and Ewald 8], and to the lecture notes [40]

(a) The linear hull, the affine hull, the positive hull and the conve hull of a set B of vectors of W, r > 1, will be denoted by lin(B), a f f ( ) pos(B) (or M>0 B and conv(£?) respectively. The dimension d i m ( ) of a W is defined to be the dimension of its affine hull

(b) Polyhedral cones. Let N = 1r be a free Z—module of rank r > 1. N can be regarded as a lattice in AR := N (g>z K = W. (For fixed identification, we shall represent the elements of Nu by column-vectors in W). If } is a Z-basi of AT, then

det(AT) := |det ( r i i , . . ,nr)\

is the lattice determinant. An n £ is called primitive if conv(0 n})flA contains no other points except 0 and n.

Let N = Zr be as above, = H (JV, Z) its dual lattice, iVR, MR their rea scalar extensions, and {,.) : iVR x MR —>• K the natural K-bilinear pairing. (For fixed identification MR = W, we analogously represent the elements of M R by row-vectors in W). A subset a of JVR is called strongly convex polyhedral cone (s.c.p.c for short), if there exist ni,...,rifc £ NR, such that a = p o s ( { r , . . . , n^}) and a fl (—<T) = {0}. Its relative interior int(cr) (resp. its relative boundary da) is the usual topological interior (resp. the usual topological boundary) of it considered as subset of lin(er) The dual cone of a is defined by

av : {x MR | (x,y) > , Vy, y € a

and satisfies: CTV + (—a) = MR and dim(erv) = r A subset r of a s.c.p cone a is called a face of a (notation: r -< er), if r = {y £ a | (mo,y) = 0 } , for some m0 £ a v . A s.cp.c a = pos({ni , . . . ,n^}) is called simplicial (resp. ra-tional) if r . . . ,r are IR-linearly independent (resp. if rii, r JVQ, where iVQ: Q)

(c) Monoids. If e C A^ is a rational s.c.p. cone, then a has 0 as its apex and the subsemigroup a fl N of N is a monoid. The following two propositions describe the fundamental properties of this monoid a fl N and their proofs go essentiall back to Gordan [25], Hilbert [28] and van der Corput [6] , [77]

Proposition 2.1 (Gordan's lemma) is finitel generated as additive seigroup e. there exist

ri2 nv € a f suchthat N = >0 r + Z>0 « Z > 0 n „ .

Proposition 2.2 (Minimal generating system). Among aiJ systems of gener ators ofaCN, thee is a system Hlbjy (cr) of minimal cardinality which is uniquel determined up t the ordering of its ents) by the followin characterizati

lbj (cr) = n £ a n (N \ })

n cannot be expressed as the sum of two other

vectors belonging to a n ( A T \ } )

] 33.

Definition 2.3. Hlbjv (a) is often called the Hilbert basis of a w.r. N.

About algorithms for the determination of Hibert bases of pointed rational cones we refer to Pottier [55], [56], Sturmfels [72] ( 3 2 p. 28) and Henk-Weismantel 27] and to the other references therein.

(d) Algebraic tori defined via N. Let C* be the multiplicative group of non­zero complex numbers. For N = Zr we define an rdimensional algebraic torus

(Cby = H o m C ) =

ig ara (m) : it

( m ) ( ) :(m),

We have

(m + ) = e (m) e (m) , for TO, TO , and 0) =

Moreover for each n G AT, we define an parameter subgroup

7„ : C T with 7 n (A) (m) := A<mn>, for A G C , TO G ,

(7n+„' = 7ra° 7 n for n ,n € AT). We can therefore identify M with the character group of TN and N with the group of 1-parameter subgroups of Tjv- If {ri n} is a Z-basis of AT and {mi , . . . ,mr} the dual basis (of ) and if we set (r) Vj < r, then there exists an isomorphism

J V 9 t A ( « 1 ) « r ( ) ) e ( C

and } plays the role of a coordinate system of . Hence to an

_, a ( resp. to an n = \] b i j i

we associate the character ("Laurent monomial") (m) = u^ u® • •

( resp. the parameter subgroup : C A A G (C ) .

On the other hand, for a rational s c p c a with

M R CTV = Z > o " Z > 0 2 I Z > o fc

we associate to the finitely generated normal monoidal C-subalgebra C[ n c of C [ ] an affine complex variety

Ua := Max-Spec (C[ n a v ] ) ,

which can be identified with the set of semigroup homomorphisms :

u : n a v 1, M (T + ( T

for all m , m n <

re ( T (U ( T Vm fl c d V , U G

Proposition 2.4 (Embedding by binomials) In the analytic category identified with its image under the injective map ( ( T O ) . . , e (m^)) : U„ * Cfc, can ded as an anaytic set determined by a syste of equations of the form:

(monomial) = (monomial) is analytic structure induced on is independent of the semigroup generators {m,.. m and each map e (m) U„ is holomor phic w.rt. it. In particular, for r -< a, UT is an open subset ofUa. Moreover, # (HlbM (CTV)) = d (< k), then d is nothing but the embedding dimension ofUa

i.e. the minimal number of generator of the axi ideal of the local Clbr ®u (oeC)-

Proof. See Oda [53] prop. 2 and 1.3 pp. 4-7

(e) Fans. A fan w.rt "Lr is a finite collection of rational s c p . cones in VR, such that :

(i) any face r of a G A belongs to A, and

(ii) for CTI c G A, the intersection <j\ fl cr2 is a face of both CTI and a^-

The union | : \ a e } is called the support of A. Furthermore, we define

A (i) : {cr £ A | dim (a) = i} , for « < r .

If ß € A( l ) , then there exists a unique primitive vector n(g) £ C\ g with Q = K>o n (g) and each cone cr € A can be therefore written as

° = Y M>o n (e) €A

The set Gen(cr) := {n (g) \ g e A (1), g -< cr} is called the set o/ minimal genera­tors (within the pure first skeleton) of a. For A itself one defines analogousl

Gen (A) := ( J Gen {a) . o-eA

(f) Toric varieties, orbits and stars. The toric variety X(N,A) associated to a fan A w.rt . the lattice N is by definition the identification space

X(N,A):=((\J UA j

with Ua ~ «2 Ua2 if and only if there is a r € A, such that r -< o\ fl CT2 and u\ = W2 within ^ (cf. lemma 3.1 below). As complex variety, X(N,A) turns out to be irreducible normal, Cohen-Macaulay and to have at most rational singularities (cf. [22], p. 76, and [53], thm. 1.4, p. 7, and cor. 39 , p. 25) X (N, A) is called simplicial if all cones of A are simplicial.

• X (N, A) admits a canonical action which extends the group multiplication of TN U{0y

X(N,A)u)^u€X(N,A) (22)

where, for u &Ua, (t • u) (m) := t (m)-u (m), Vm, m G MHCTV . The orbits w.r.t the action (2.2) are parametrized by the set of all the cones belonging to A. For a T G A, we denote by orb(r) (resp. by V (r)) the orbit (resp. the closure of the orbit) which is associated to r . The spaces orb(r) and V (r) have the following properties (cf. [22] pp. 5255 [53] 3) :

(i) For r € A, it is

V (r) = orb (CT) | C T G A , T -

and orb(r) = F ( r ) \ V (a) | r ^ C } .

(ii) If r G A, then V (r) = X (AT ( r ) , Star (r; A)) is itself a toric variety w.r.t.

N(T):=N/ := n lin (r) , Star (r; A) := {a \ a A, r -« a} ,

where ö7 = (<r + (NT)R) / {NT)R denotes the image of a in N (r)R R / (i

(iii) For r G A, the closure V (r) is equipped with an affine open covering

[T) \r^a

consisting of "intermediate" subvarieties

UT (r) = orb (r) Ua (T) £7ff

being defined by : ^ (r) := Max-Spec(C[ (r)]) with (r) denoting the dual of JV(r)

(g) Smoothness and compactness criterion. Let JV = be a lattice of rank r and CT C iVu a simplicial rational s.c.pc. of dimension k <r a can be obviously written as CT = for distinct dimensional cones g g. We denote by

ar (CT) : y G (N y = , with j < Vj

the fundamental {half-open) parallelotope which is associated to CT. The multilic­ity mult(o-; N) of CT with respect to is defined as

mult (CT JV) : (Par (CT) Na) = Vol ( a r (CT) ; ) ,

where Vol ( a r (CT)) denotes the usual volume of ar (CT) and

its the relative volume w.rt

Proposition 2 5 Th e toric variety U„ is smooth iff m l t ( a ; N Cor respondingly, an arbit toric vaety X (iV, A) is smooth and l it is silicia and each one a satisfies this onditi

Proof. It follows from [53], thm. 1.0, p. 5 •

• For the systematic study of toric singularities it is useful to introduce the notion of the "splitting codimension" of the closed point orb(cr) of an Ua. For the germ (JJa, orb (a)) of an affine r-dimensional toric variety w.r.t. a singular point orb(cr) the splitting codimension splcod(orb (er); ) of orb(er) i is denned as :

splco (orb (e = max r a =* U x ", dim (er') = a and

Sing {Ua) ^ 0

If splcod(orb (er); Ua) = r, then orb(er) will be called a c-inlrity i . a singularity having the maximum splitting codimension.

• Next theorem gives a necessary and sufficient condition for X (N, A) to be com­pact

Theorem 2.6. A oric vaety X (N, A) is compact i and l is a complete fan, , |A| =

Proof. See Oda [53] thm. 11 p 16

(h) Order functions, support functions and divisors. If X (JV, A) is a (not neces sarily compact) toric variety associated to a fan A w.r.t. a lattice N, M its dual and L : Tjy "^ X (N, A) the canonical inclusion, then t* (OTN ) is a Tjy-invariant quasi-coherent sheaf of öjc(7v,A)-modules canonically embedded into the constant sheaf C (X (N, A)) of rational functions of X (N, A). Let T ^ 0 be a TV-invariant coherent sheaf of fractional ideals over X (N, A) contained in t* (OTN). Fix a s c p cone a € A, n € N fl er, and consider the corresponding parameter group3:

7„ : Spec [w '1]) = C = Spec (C [ f ] )

Since 3 lim A) e (cf. [53] (v) p. ) is extendable to a map A -

7 ~ : Spec (C[ ] ) = M U« = Spec (C[M n a]) .

The coherence of J7 implies that ^|c/CT is of type J (cf. [6] pp. 1 1 1 1 1 ) with being an graded complex vectorsubspace of

OTN)=C[]= 0 (m)

3 Here we drop the prefix Max- because we do not work only with closed points but a l o wi closed subsets, and the Cscheme structure is essential for the arguments.

i t l n e ] l

C[ n av] , for some \ m and q £ N,

on the other. The pullback ~%^ is realized via the finitely generated C [ ] module

^ C H w K " ) C()

Define the orer-function w.rt JF by

ord^r (n) : inf < ( n qa 6 Z .

This ord (n) is exactly the image ordo (^" U ) under the usual order function

ord : C ( >

of the discrete valuation ring ö ^ 0 with «; as uniformizing parameter. Since the above definition depends only on er, one extends ord^- to the entire | | by setting

ord (y) : inf { ( y ) Vy, y £ a, a £ A .

The order function ordjr is M-valued and has the following characteristic properties:

(i) it is positively homogeneous, i.e. ord^r (cy) = c ordjr (y) for all c £ R>o

(ii) ordjr |„ is piecewise linear on each a £ A,

(iii) ord^-(JVn|A|) c Z, and

(iv) for all c £ A, ord^- \a is «pper convex i e

ord | (y y ) > ordjr \ (y) ord ( y ) , for any pai y, y £ a.

Definition 2.7. Let X (JV, A) be a toric variety. A function ip : | A| — E is called integral PLsupport function if it satisfies the above properties (i)(iv) We define

PL-SF (JV A) • ^ i n t e S r a l P L-support 1 ^ > ' ' | functions defined on | | J

and the sets of integral (A-linear) suppor functions

4 An integral A-linear support function ip is called strictly upper convex if it is upper convex on | A and if for any two dijoint maximal-dimensional cones a and a', the linear functions ma

mai e M = om z ( iV : Z) C omR ( N R M) = M R denning ip | = (m*) and ip \i = < / , » ) are different.

(J (N lin

u

u SUCSF (N, A) : U G UCSF (JV, A) ^ S t n c t l y U ^ r

v ' yT \ ' / convex on | J Analogously, one defines the sets SFQ (AT, A), U C S F Q (N, A), SUCSFQ (AT, A) of rational support functions by modifying property (iii) into : TP(NQC\ | Q. All the above sets are equipped with the usual additive group structure Theorem 2.8. For functi if) G PL-SF(AT, A), and an arbit one a G define

W M l<™.y>>^(y) Vy, y e a } . GM

The family of Tinvariant sheaves (J^) | a G A } being assciated to the am­ly of ideals {(Jp)a | a g A can b glued together (cf. [6] , ExII.1.22, p. 69)

construct a coherent sheaf T$ ofinvariant fractiona idea ver X (N, A) ontained in t ÖTN) Morver

(i) o r d ^ ip,

(ii) .Ford.p is the completion of T (in Zariski's sence);

(iii) mappin ord and one obtains a ijecti

, , coherent sheaves of Tjv-invariant ^ ' ' i: | complete fractional ideals over X (N, A)

(iv) ^ i ö ord^F > ij), o r d ^ . ^ = ord^1 ord^ and

(v) F^,1 = !F as OX(Nmodue sheaves) < 1 is linear

Proof. See SaintDonat [40], ch. I, § 2 thm. 9 pp. 283 D

Definition 2.9. Let X (N, A) be a toric variety and a G A. The convex interpo­lation ip^ of a function $ : Gen(a) Z is defined by

G M and (m, y) > i? (n (<?))

ß ß G A (1), with -< c J

Correspondingly, by the convex interpolation of a real function $ : Gen(A) —> Z is meant a function tp$ : | A| — R, such that ^ |CT is the convex interpolation of i? |CT

(in the above sence) for all a G A. Obviously, such a V># belongs to PL-SF(A/', A) and conversely each integral PL-support function has this form.

• Let X be an g WD correspondence

T_X (X) } A Ox (D)} €

reflexive coherent (ip. torsion-free) sheaves

of fractional ideals over X having rank one

ith x (D enned by sen in every non-emty open subset of X nto

^Ox(D)():={<peC(X) \(div(ip)+D)\ > 0

induces a Z-module isomorphism (cf. Reid [58], App. to ); in fact, to avoid torsion, one defines this Z-module structure by setting:

Ö ( £ ! + D) := (Ox ( A ) ® Ox (D)) , 8 (K D) := Ox (D)K] =OX(KD)

for any D,DUD2 e WDiv(X) and K G 1.

For X = X (N, A) any rdimensional toric variety let now

VWDiv (X) and w C D i v (X) = WDiv (X) n CDiv (X)

denote the groups of Tjy-invariant Weil and Cartier divisors respectively. If is not contained in any proper subspace of NR, then6

i c X ) r-1 X )

i X) WDi X)

is commutative diagram ith exact columns TjyWDivX) has as basis:

VWDiv (X) = V (Q) \߀A ( } .

5 Reflexive coherent sheaves T are those which are isomorphic to their biduals Ty v (wher JFV := Homox (F, Ox))- For T C C ( X ) of rank one they are also called divisorial.

6In particular, for X smooth and compact, Pic(X) is torsion f e e and the Picard number equals # (1) — r.

T h o r e m 2 1 i v i s r s nd s u p p r t functions xis respondences

. , evaluation functions PL-SF(iV,A) tf:Gen(A)^

t reflexive coherent (i.p. torsion-free)

sheaves of rank 1 of WDiv (X) invariant, complete

fractional ideals over X = X (JV, A)

in fact Z-modu isomorphis) induced by mappin

^ > V" = ^ — > D = D

and D = D = Ox (D) o r d ^

with as above and

6 A i )

M o r v e r

D4,€ Tiv-CDiv (X) , n (r),. • L s ^ V> = ord^GSF(JV,A). (ie = Ox (D) is mvertible) J

Proof. See SaintDonat [ ] ch. I, § 2, thm. 9, pp. 2831.

Theorem 2.11 ( i°-generated) If X = X(N,A) is a compact toric vaet andip G SF(iV,A) then

Ox (Dip) is generated by its global sections -<= UCSF (JV, A)

Proof. See Oda [53] thm. 27 p. 7 •

Theorem 2.12 (Ampelness). Let X = X (JV, A) t necessail ompact oric variety. Then a divisor

D = D^ CDiv (X) (resp. D = D CDiv (X) ® Q)

is ampl f and and only if

i> G SUCSF (JV, A) (resp. $ G SUCSFQ (JV, A))

Proof. See Kempf [0] ch. I § 3 thm. 3 p. 48

Corol lay 2.13 (Quasiprojectivity). Let X (AT, A) b a toric variety (resp. compact oric variety). Then X (N, A) is quasiprojective resp rojective and

SUCSF (N, A) ^ (or equivalent^ SUCSF (AT, A) ^ 0) .

(i) Intersection numbers of Cartier divisors on toric varieties. If X is a normal complex variety of dimension r, and D\,...,Dr Cartier divisors on X, such that W := fli=i ( s u P P ( ) ) is compact, then their intersection number is defined to be the degree

( £ • • ) : degjy ( • • }) e

of the zero-cycle

D 2 • • } : (D2 • • } G A 0 (W

determined inductively as usual (i.e. probably after passing to the corresponding pseudodivisors) See Fulton [21], Ch. 1-2; in particular, 1.1.4, p. 3, and pp. 38-39. For X = X (N, A) a smooth toric variety D V () for all i < r and s pairwise distinct rays we have

, if g1 H / N

( D - - ) = ' yi . (23) , otherwise

Moreover one obtains by general techniques:

e m m a 2.14. Let X (N, A) be an rdiensional smooth toric variet and

V(Q1)Dr V(ß

divisors on X. Suppose that either X itself or at east is compact and that 0i 02 > wii all the ther rys are distinct. Then

(i) IfT:=Q + h Qr A, the intersection number (D • • • D vanishes.

(ii) Ifr € A, there exist g A ( and i n t e , K such that

n{Qn{ r£A(r) r € A (r)

In ti is case • • • ) = ( 2 4 )

Proof. See Oda [53], p 8

There are also easy generalizations for A simplicial but we shall not use them because we shall work exclusively with intersection numbers of divisors on smooth X (N, A)'s. Another method for the evaluation of intersection numbers is based on mixed polytope volumes

roposition 2.15. Let X (iV, A) e an r-dimensional mpact ric variet UCSF(iV, A) then the selfintersection number of D equa

= ( r ! V o l ( (25)

whe := PD = {X G M R I <x n (g)) >^{n (g)) }

is the lattice polytope associated to the divisor D = D^ defined in 2 . Mor geneally for r uppe onvex functions .. one ha

• • J = (r!) V o l , ^ (2

whee Vol {P1pl, • $r) dentes the ixed vol of the polopes

Proof. See Oda [53] prop. 2 p. 79.

(j) Euler-Poincare characteristic. The topological Euler-Poincare characteristic of a (not necessarily compact) toric variety can be easily read off from the maximal cones of the defining fan

Proposition 2.16. Let X (iV, A) be an r-dimensiona oric variet assciated t A. Then the tological Euer-Poincare characteristic

2r

(X (N, A)) = dim {X (N, A) ; Q)

of X (N, A) is equa the number of r-diensiona ones

X(X(iV,A)) = (A(r)) (27)

Proof. See Fulton [22], p. 59. D

(k) Maps o fans. A map of fans TO : (JV, A ) — (iV, A) is a Z-linear homomor phism w whose scalar extension w : J —>• NR satisfies the property:

V C , a 3 a, a G with TO (a1) C a .

ro®zidc* : TN, = N' ®z > N (gz C* is a homomorphism from to and the scalar extension TOV : MR —»• MR of the dual Z-linear map TOV : M —• M' induces an equivariant holomorhic map TO* : X(NA) —• X(N,A) as follows: If TO (a') C er for a G A, a then obviously T n crv) C R (c and the holomorphic map

^, —• J7CT with (u) (m) : (TOV (m)) Vm, m <7V,

is equivariant because

TO* ( ) (m) = ( ') (TO (m)) =

( T (m)) M' ( T (m)) = TO* () (m) • TO* (U') (m)

for all t' G Tjv, m G M R CTV . After gluing together the affine charts of A and of we determine a welldefined map TO* : X (i ) ->• X (AT, A)

Theorem .17 ( P r o r n e s s ) . If t (N', A') —>• (NA is a map f fan m* is proper if and on if w~x (|A|) = | A | . In particuar N = and ' is

refinement of A, e. if each one of A is a union of ones of A then the olomorphic ap id : X (N, ) > X (N, A) is proper and birational

Proof. See Oda [53] thm. 5 p. 2 and cor 8 p. 23

3. Blow-ups and resolutions of toric varieties

One of the most fundamental cornerstones of various significant constructions of birational morphisms between complex varieties is the blowing up along subvari eties or -more general- along closed subschemes.

(a) Local construction. Let U = Max-Spec(i?) be (the closed point set of) an affine noetherian scheme, I an ideal of R, Z = Max-Spec(i?/J) and S (R, I) := © d > 0 I The homogeneous spectrum Proj(Sl (R, I)) of S (R, I) together with the structure morphism

Bl7 (R) := Bl£ (U) := Proj (S(R,I)) - A U

is called the blow-up of U w.r.t. i" or the blow-up of U along Z (or of with center Z). If •, hß} is a set of generators of / , then

BV ) = Max-Spec I R 0

with R \j^ -j^\ viewed as an i?-subalgebra of R^. The exceptional locus of

ß is Exc(/3) := / _ 1 (Z) and its contraction locus = Z. Moreover, ß~l (I~) is invertible (with J~ denoting here the sheaf being associated to the ideal i", and with i" regarded as an Ä-module, cf. [ ] p 110) and /?"1 ( I ) ^ S (R, I) (

(b) Globalization by gluing lemma. Let {Xj \ j G J} be a family of schemes

Lemma 3.1 (Gluing schemes). If there exists a collecti {X \ j } of open sets k of and isomorphis of schees

kj k0X j> k,Ox kJ

satisfyin the conditions

• X jt and = id

• V k , = idXji and

* (rh,k ° Vj,i) k,*,4 = ,,*,« whe Xjkj Xjik ^ ) , then there exists a sche an open ve {Wj \ j e } ofW, and a collecti

of isomorphis < f | j G such that kj = f ° fj1 j for all

j € J

The scheme W obtained by gluing the me f a i l € J ia the above isomorphisms will be denoted by

II or s imPly by

if the gluing isomorphisms are selfevident from the context

Proof. See e.g. [26], x. II.2.12, p. 80. •

Let now X be a coherent (non-zero) sheaf of Ox-ideals over a complex variety X and

{Uj Max-Spec (Ej) | j e

an affine cover of X. For every J, we have Z f° r some ideal of (cf. [6] I I54 ) . Considering

{ßr. Bl c \j£j}

as in (a) where Zj = Max-Spec(Rjfl) and Z the closed subscheme of X denned by

Z = suW(Ox/l) : {x e X | (Oxfflx ^ } ,

we determine natural isomorphisms

j 1 A i1 ) .

Applying lemma 3.1 to the family {ßj1 ) | j G we construct a birational proper surjective morphism n = TTX

) : ) = roj 0

as the natural projection induced by Ox —> 1 ® > o d> w ^ n r o J denoting

the <//o&aZ homogeneous spectrum (as in [26], p. 160). [ B l z (X) ,TT\ is £fte blow-up

of X a/on<? I or the monomial transformation w.rt . I (with center Z). Let us recall its main properties :

7T induces an isomorphism ~B\Z (X) \ 7r_1 (Z) ^ X \ Z, ie . Exc(7r) = 7r_1 (Z).

The algebraic scheme B l z (X) is a complex variety (cf. [26] II 716 (a) p. 166)

If both X and Z are smooth, then B l z (X) is smooth too

The preimage sheaf of I, ir~lT Ox '•= Im X — 0 B I is mvertible and

determin c(7r). Hence, Exc(7r) represents a Cartier (not e c e r i l y prime) divisor of B X) which is isomorphic to the projectivization F NC X)) of the normal cone

NC X) : Spec 0 >0

of X along Z. The relation between X) and Exc(7r) is described by the isomorphisms

• ^ E f ( X ° B l f x c ( 7 r ) ) EXC(TT) ^ ( N C ) '

In particular, if Z is a local complete intersection in X, the canonical epimorphism from the d-th symmetrizer sheaf of I / 1 2 onto the d-th part of the normal cone graded algebra

Sym ( X / ) -

becomes an isomorphism and therefore

xc (TT) roj 0 Sym ( l / ( X / ) = P ^ d>o

(Here A( " v denote the corresponding normal and conormal sheaves.)

(c) Universal property of blowing up. If g : X' —> X is any morphism, and Z (resp. Z') is defined by the ideal sheaf X (resp. by X' := 7r_ 1I • 0j>) , where •K denotes the blow-up-morphism of X along Z, then composing the morphism Bl z , (X') — (X) with the projection to X' , we get the commutative diagram:

f (X) t

-> f (X) x X —>

If g~ TÖx' is invertible, then it is easy to deduce an isomorhism B ( X ) =

Hence, there is a unique morphism h factorizing g. This means that ( B (X)

is universal among all pairs (Xg) having invertible ideal sheaves _ 1X Ox

(d) Blowing up intermediate subschemes. Let Z C x, W C x be two closed subschemes of X which are defined by the ideal sheaves X and J respectively such that Z n W = swpp(Ox + J) ^ 0 and n is nowhere dense in and let

TT = T (X ) X , TT | e s : f + ^

denote the corresponding blow-ups. Then, by the above mentio the morphism TT, one verifies easily the following isomorphism:

(X

schemetheoretic closure of the preimage ^ restr

n - ' i ^ i Z ) ) withi X) ^

This closure is called the stric transform S R ( W , n) of W, n) uner If we assume that W <£ Z and Z then S T Z , X, T n S T l 7 , 7 ) = 0 (cf

6] x. 7 2 p 71)

(e) Normalization process. Even if the complex variety X itself is normal and Z = supp(£>x /X) smooth, with Zf~\ Sing(X) ^ 0 , (X) is noi necessarily normal Using an affine cover of X

j Max-Spec(R) | j G } and , = Max-Spec(Rj/I

(as i b)) and the (finite) normalization morphisms onto Bl ) :

: Norm ffl —• Bl ) ,

we define the normalized blow-up (Norm B l z (X) ,7Tx of X w.rt Z (or

with center Z) by patching the affine pieces together :

he combination of the universal roperty of the normalization morphism v% (see [26], Ex. 3.8, p. 91) with that of 7rx (cf. (c)) leads to the universal property of normalized blowing up: If g : X —> X is any proper morphism, X' normal and g~xX Ox is invertible then there exists a unique morphism h factorizing g :

Norm X) ^

X

(f) Some warnings. To blow up arbitrary subschemes Z of X requires great care

Sometimes for different coherent ideal Ox-sheaves X, X over X and

= supp (Ox IX) = supp (Ox / ) ,

it is possible to have an isomorphism lf (X) = Blf, (X). For instance, for X an arbitrary coherent sheaf of ideals and X' an invertible sheaf of ideals, X and X • X give isomorphic blow-ups. The same remains true for X and with d > 2 (see

6], Ex. 7.11, p. 171)

On the other hand, if Z is endowed with two different scheme structures

Z = Supp(Ox/X) = supp(Ox/J), J,

then B X) and X) are in general too far from being isomorphic to each other

Example 3.2. Perhaps the simplest example is to consider X = A . to be the affine complex plane with coordinate system {x,}, Z = { } the zero point X = (x, öß the maximal ideal of £ ^ a n d = x, 0$ Then

, } (

is the usual blow-up with exceptional set

^ ( { } ) ^ and S i n g ( O ) O ( ) ) = 0 .

In contrast to the usual blow-up case and although the blow-up

B{o,o)} ^c ^

w.r.t. the prime ideal J has also a smooth rational curve as exceptional set the singular locus of Bl | /0 0-.y (A|.) is non-empty. As one may easily verify, it consists of a single ordinary double point which lies on this rational curve. The real parts TT^1 (£)) and J1 (33) over a small disk J) A | c A . centered at (0,0 are illustrated in figures 1 (a) and 2. The real part of ir^1 (23) is used to be viewed as a spiral staircase (whose stairs extend in both directions). Away from the origin we get an isomorphism, and the points of xc(7Tx) are in 1-1 correspondence with the set of straight lines passing through (0,0) (see fig.l (a)). One may, of course think of it just as an enlargement (Aufllasung) of the origin spreading out the "tangent directions" (see figl (b)). In fact, it can be shown that the topological space of the real part of JTJ1 (23) is homeomorphic to a Möbius strip, while, in the analogous setting, TT~^ (23) has a marked "twisted" point which corresponds to the occuring singularity (see fig). In both cases the dotted meridian line indicates the exceptional set.

Vpi

ig (a

ig (b

25

ig

• The close subsche Z of X can be in general equipped with lots of scheme structures Oz, with Oz = Ox / I for different X's. There is, however, a unique scheme structure above them, which is the "smallest one" ; namely, the reduced induced structure Oz / 9^z (supporting the same underlying topological space) where 3Kz is the O^-ideal sheaf defined by

{open sets of Z 3 9\z {U) : the nilradical of Oz (

(see [26], II.3.2.6, p. 8 ) . Let ZTed = supp(0z/$Hz) denote the reduced sub-scheme associated to Z. The singular locus of Z can be written as

Sing (Z) = Sing ( Z r d ) non-reduced points of Z} .

From now on, the uniquely determined blow-up B z| z ' z (X) of X along this

reduced subscheme Zd will be denoted simply by (X) and will be called the usual blow-up of X along Z.

Another notable property of a blow-up is the projectivity of its defining morphism (and consequently the projectivity of because is finite)

7Since we allow Z to carry a non-reduced structure (in contrast to X which is assumed to be a complex variety, and therefore always reduced), Sing(Z) might become very "big". In particular,

in the above example 3.2, for Z = { (00 )} suppl Ö / J , we obtain Z = ing(Z)!

Proposition 3.3. If is quasiprojective (esp rojective) then both

and Norm B X) will e quasprojective resp projective as ell

Proof. See Hartshorne [6] , prop. I I 7 1 6 p 166. D

Corollary 3.4. Let X be a quasiprojective (resp. projective) complex variet and Z = supp(0x 11) a closed susche of If X0 : X, Z0 := Z, X0 := X and

{ _ , : X - (*j-i) -i

is a finite sequence of blowups with = supp then is quasiro­jective resp projective)

• The troubles with the projectivity begin whenever one wishes to blow up (once or more times) "overlapping one another pieces" U\,...,Uk of a fixed closed (possi bly reducible non-reduced or singular) subscheme Z of a quasiprojective complex variety with (ji=1 Ui, and try to glue the blown up (non-disjoint) overly­ing parts, say W\,..., Wk, together. Even if this gluing procedure is absolutel natural the resulting morphism

f : X ] l ^

(with contraction locus Z) is not always projective. The two classical examples of this kind, having as starting-point two smooth curves meeting trasversally at two points and a curve with an ordinary double point on a smooth complete threefold respectively, are due to Hironaka. See Hartshorne [26], pp. 443-445, for the details of the construction. The non-projectivity of such an / implies automatically not only the non-quasiprojectivity of X' but also the fact that / cannot be expressed as a composite of finitely many blow-ups (by cor. 34) . For another simple exam­ple see rem. 75 below.

(g) Desingularization by successive blow-ups with smooth centers. We just for­mulate here the famous theorem of Hironaka which guarantees the existence of (full) desingularizations by performing a finite number of monoidal transforma tions with smooth centers. (Of course, we should again stress, that not ever desingularization can be composed of finitely many blow-ups

Theorem 3.5. (Hironaka's Theorem8 , [29]). Let e an comlex variety Then the exists alw a unite sequence of blowups

{ ^ : X - -i _,

such that X0 = an S i n g ( ) = 0 , whe is a smooth suvaet of S i n g ( J ) V j , 0

8 The original result of Hironaka is more general. It is valid for any equicharacteristic zero excellent scheme X. The centers Zj are normally flat i the ambient space.

(There are meanwhile consideable simplifications of the original proof of thm 3.5, like those due to Bierstone & Milman [5], [6], who introduced an appropriate discrete local invariant for points of X whose maximum locus determines a center of blow-up leading to constructive desingularization)

(h) Toric blow-ups. Working within the category of toric varieties, both blow­up and resolution are much more easier as they can be translated into purel combinatorial operations involving specific cones of our fans.

Theorem 3.6 (Toric normalized b l o w u p ) . Let X (JV, A) be a toric variety / JVinvariant coheent ideal shea defining a subschem of X (N, A)

with T = (0XN,A /Z)\z ontained in ÖTN) (whe L : X (JV, A) dentes the canonica injecti) and

Norm [ B I | (X (JV, A))] X (JV, A)

the normalized blow-up of X (JV, A) along Z. Then TTX ° vx is the Tj^-equivariant olomorphic ap and the overlyin space is identified with the toric vaet

Norm (JV, A)) X (JV, A (I ; Z))

entes the follwing refment A :

the fan in JVR defined by rational s.c.p. cones which constitute the maximal subdivision

_. (w.r.t. usual inclusion) of the cones of A, ' • go that the restriction of ord (J7)

on each of them becomes an integral linear support function

Proof. Since ; Z) is a refinement of the initial fan A, the toric map

id* : X (JV, A u ( I ; Z)) —• X (JV, A)

is a proper birational morphism by thm. 2.17. Since T = (OX(N,A) /%) \z is invariant, TV acts on B l z (X (JV, A)) too. T is invertible over the open set of X (JV, A). Hence the normalized blow-up is an isomorphism over

V (X(JV,A))

is an equivariant immersion and itx ° vx a torus-equivariant map. Combining the universal property of the normalized blow-up (see (e)) with the fact, that for any a e A, the inverse image of X restricted onto Ua is an invertible sheaf if and only if ordjr \a is linear (by thm. 2 ) we deduce that id* = -KX°X (up to isomorphism overX(JV,A))

7 (i) I X (J is th,

A ( ) = Q}i Gen (A) = nk},

with m=n (ß j for

0 := n nk

and

p o s ( n n _ i n for

pos(n0ni,.nk-i for j =

then, fixing a cone r € A, every cone CT € A with r -< er can be written as

CT = + , for some cone A, with r = } .

Setting for all j and

» : = ( A x C T e ^ C } ) f a c e S o f ^ J ^ J

we get X(JV,A*(T)) = Vf r ) X(iV,A))

with A* (r) = Abi (X; V (r)) the fan "starring" r where X is now the usual ideal (with no nilpotent elements) defining the closure V (r) as subvariety of X (N, A) (cf. Oda [53], prop. 1.26, pp. 38-39; see also Ewald [18], § VI.7, for an equivalent combinatorial characterization in terms of "stellar subdivisions".)

(ii) If X (N, A) is not smooth, then performing the above starring subdivision w.r.t. r, we get a normalized blow-up of X (N, A) along V (T), with V (r) being equipped with a not necessarily reduced scheme structure! For a simple example see 72 below.

(iii) Even if X (N, A) is smooth, applying theorem 3.6 for a nonreduced subscheme

Z as center, the resulting normal complex variety Norm p(JV,A)) ] is not

necessarily smooth. The simplest example is to take lei © Z e , i e , the standard lattice with the unit vectors M Le Ze^ its dual

CT = pos(ei ,e 2 ) , ?7 C =C Z = ( , 0 ) = orb (CT) , and X = ( e ( e ) ( 2 e ) )

Then the blown up space corresponds to the fan consisting of the cones

I = pos((2 ) T , e ) , p o s ( e ( 2 , ) T ) ,

together with their faces. Obviously, mult(<ri;iV) = 2, and we rediscover the second example of 3.2. (In fact, this is nothing but the so-called weighte blow-up of the origin w.rt . (2 i Reid's terminology; see [58] p. 297)

(iv) On the other hand, the nomalization of the ual low-up of a not necessaril smooth, affine toric variety Ua = Max-Spec(C[M n erv]) along a subvariety can be described intrinsically by making use of thm. 3.6 and arguments coming from an embedding. Next proposition treats of the case in which one blows up Ua at orb (c)

Proposition 3.8 (Usua n o r m l i z e d blow-up at the closed point).

Let lattice of rank r, a an rdiensional rationa one in NR and

U = Max-Spec (C[M D CTV])

the assciated aine toric variety with = Homz (JV, Z) Morver let

lb ( c ) = m i m } , r d

denote a fixed enumrati of the emb of the Hilb asis of w.r (cf. prop. 22). Then

Norm Ua X (N, A orb (a)}) ,

whe

i i M (( maximal . , A 7- 11 < j < d} together orb (a)] : {( . ^ ; orb (a) ' J . ^ " ^ ^

with

j y £ | (mimy) > , M i G 1,. j , + 1 d}} ,

for all j , 1 < j < d. (Warning. Though the union of the above er/s forms always a fan, it might happen that Oj ^ or that Cj is a proper face of ay, for certain distinct indices j , j ' belonging to { 1 , . . . , d}. In this case, we just ignore the superfluous cones and introduce a new index-enumeration for the rest preferabl by considering only the maximaldimensional ones.)

Proo Under the above assumptions, we may use prop. 2.4 and embed Ua by ( m ) , ( m ) ) "minimally" into the affine complex space

= Max-Spec c [ M

where

p o s ( i , m } ) C J Zn • • Zm

orb(cr) is mapped onto 0 = orb(a) € C, with

a = (N , N = Bomz C l ^

Using the embedding N •—>• N, it is possible to describe Bl^^o-) (Ua) as the strict transform of Ua under the usual blow-up morphism of C at the origin (ie. just by applying what we mentioned in (d) for Z = orb(er), W = Ua and X = Cd). Hence we obtain the following commutative diagrams of torusequivariant holomorphic maps:

Norm Ua

E x c ( 7 r | e s ) Y0l1 [Ua XC(TT)

4- w Ies orb (a) G Ua > C 3 0

Since Bljje (Cd) is realized as toric variety by the d cones of the barycentric subdivision of a,

:= {y G | ( w m ^ y ) > , Vi, G . , - . ,d}} , 1 j < d

(cf. 3.7 (i)), the above defined er/s are exactly the restrictions of TS on a and determine obviousl orb (a)] as it was given initially in thm. 3

(i) Resolutions of toric singularities. To resolve toric singularities is actually equiv­alent to subdividing simplicial cones into others of smaller multiplicity

Theo rem 3.9 (Resolution of toric singularities. Weak version).

If X (N, A) is an arbitrary toric vaiety then there exists a refineent of A, such that

f = id*:X (N, A X (JV, A)

is a full) desingrizati of X (N, A)

Sketch of proof. Considering the multiplicity of a simplicial cone a as a volume and using the wellbehaved volume properties under subdivisions w.r.t. lattice points of Pa r (a) one can easily desingularize equivariantly any toric variety X(JV,A):

• At first we refine the cones of A in order to make it simplicial. That this is always possible follows basically from Caratheodory's theorem concerning convex polyhedral cones (cf. [69] p. 94; for a simple proof see also G .wa ld 8] III 2 p. 75, and V 4.2, p. 158).

• In the second step this new simplicial A will be subdivided further into subcones of strictly smaller multiplicities than those of the cones of the starting-point. After finitely many subdivisions of this kind one can construct a refinement A' of A, so that / = id* gives rise to a resolution of singularities of X (N,A) (by thm. 2 and prop. 25). •

If fact, for toric varieties, a single normalized blow-up of a suitable ideal sheaf is able to provide a full resolution

Theorem 3.10 (Resolution f ric singularities. S t o n g version) If X (N, A) is an arbitrary toric vaety then the exists ainvariant oherent sheaf of idea with

Sing (X (N, A)) = supp

such that

Norm g (X (iV, A)) ^ X (JV, A)

forms a (full, projective) desingurizati of X (N, A). (I is in general uniquel determined by this prope and ght contain nilptent eents)

Proof. See SaintDonat [ ] thm. 11 pp. 3235 and Brylinski [11] pp. 273-279

. Toric P f-bundles over projective spaces

This section is a brief excursus to a part of the theory of toric bundles over pro jective spaces which will be used later on (in § 8) for the precise description of the exceptional prime divisors occuring in our desingularizations

(a) An equivariant holomorphic map m* : X (N, A) —> X (N, A') of toric vari eties, induced by a map of fans w : (N, A) —> (N1, A ) (cf. §2 (k)), can be viewed as the projection map of an equivariant fiber bundle (toric bundle) over X (N, with typical fiber X (AT", A") AT" = Ker(w : AT ->• N'), if and only if w : N -)• N is surjective and the cones of A are representable as "joins" of the cones of a fan

(| A"| c A/g) with those of another fan A c A, so that the supports and

A'| are homeomorphic to each other. (See Oda [53], prop. 1.33, p. 58, and Ewald [18], thm. VI.6.7, p. 246). In the case in which its total space is assumed to be smooth and compact, this criterion can be considerably simplified by means of the notion of "primitive collections" introduced by Batyrev [ ]

Definition 4.1. If X (N,A) is an rdimensional smooth, compact toric variety then a non-empty subset 91 = {ni , r i2 , . . . ,n^} of Gen(A), k > 2, is defined to be a primitive collection if it satisfies anyone of the following equivalent conditions :

(i) For each m £ Üt, 1 < < k, one has 9? \ {rii} = Gen(o-i), for some &i belong ing to A (k — 1), while 9 itself cannot be the set of minimal generators of any

dimensional cone of A.

(ii) For each subset of indices {ji,---,jq} C { l , . . . , f c} , l < q < k, the set {nj1,..., njq} coincides with the set of minimal generators of a g-dimensional cone of A, while 91 itself cannot be the set of minimal generators of any dimensional cone of A.

Proposition 4.2 (Characterization toric bundles) . LetX (N, A) be a smooth compact toric vaety of diens and ositive inte <r.X (N, A)

is the ota space f a ric T^-bun over a smooth (rdiensona ric va riet and there exists a pitive collection

91 = nk Gen(A) ,

such that

(i) r h + = O , and

(ii) 0 , for aJJ primitive coilections C Gen(A) with ^ .

Definition 4.3. Let X(JV, A) be a smooth, compact toric variety. The fan A is called a splitting fan if any two different primitive collections within Gen(A) have no common elements

Theorem 4.4. If is a splitting fan, then X (N, A) is a projectivization of a decomosable und ve oric vaet eing assciated a splitting fan of

ll dimens

Proof. See Batyrev [1], thm. 4 3 p. 577

(b) The projectivized decomposable bundles over projective spaces, having only twisted hyperplane bundles as summands, can be easily described as toric bun­dles in terms of splitting fans with exactly two disjoint primitive collections. In particular applying 4 2 and 4 4 we obtain:

emma 4.5. Let and s positive i n t e , ( , A 2 , . . , Xk) k-up of onnegative integers and N(r-,M) (resP a attice of rank r = esp of ranks) generated by (resp. by whe

91: n n k } , n n

with the two rlations

\ + H rik rik = 0 , n ns+ Adrift .

If we define the rdiensional smooth ompact toric vaet

Y (r; A Afe) : r. r-M

by eans of the fan

r;

p o s ( ( 9 1 9 t ' ) \ { n i , n ) ,

for all (i,j) e {l , . . . , fc + l } ,

together with all their faces

then it is isphic th ta space th bun

Y (r; 5 A2 k) O © ( A ) © • •

ver r X (N A ) with

p o s ( W \ { n ; } ) in (JVJ)

f o r a J l j € { l , . . . , s + l } ,

together with all their faces

Proo By construction, 9 and 9 t ' are the only primitive collections within Gen(A(r;x1)...>xfc)) and 9?H 0 . Hence, A ^ . ^ ^ ^ ^ is a splitting fan. Since # (Gen (A'J) = # (Ag (1)) = s + 1, the basespace of the smooth toric P^-bundle Y (r; A i , , Aft) has to be isomorphic to Pp. Finally, the decomposable bundle over X (N's, A ) (whose projectivization gives the total space Y (r; A i , . . . , A*)) is isomorphic to Opj © Op« (Ai) © • • • © Opg (Aft) because the fan corresponding to the typical fiber consists of cones which are the images of the cones of under the linear map

sending a y to (y h Xknk nk)). D

Theorem 4.6 Classification theorem of Kleinschmidt) Evey smooth, compact r-dimensiona toric vaety (r > 2) with Picad numb 2 is isomorphic to one of the vaeties Y ( r ; . . A )

Proof. See [42] § 2 pp . 2 5 2 6 1

Example 4.7 (Hirzebruch s u r c e s ) Setting A = , and r = 2, one gets the rational scrolls

= Y ( 2 ; A ) A )

over Pj,, i.e. the so-called Hirzebruch surfaces whose topological, analytic and birational properties were studied in the early fifties in [30]. (Certain birational properties of them were already investigated by Segre and Del Pezzo around the end of the last century in connection with other types of scrolls and ruled surfaces See e.g. Segre [70]). It is well-known that all FA's, A ^ 1, together with P^, exhaust the class of all minimal, smooth, rational, projective complex surfaces and that can be considered as the zero-set :

{([z0:z1:z][1,t2})ercxF1c | Zl t$ z2 t$ = 0 } (4

F0 is therefore P^ Pj is isomorphic to the usual blow-up of P^ at a ( (2 ;A)

t, a d f r A 2 there exists a natural A-sheeted ramified covering FA ->• F over Fi (see [30], p. 82). The differential-topological and diffeo-morphism classification theory, and the deformation theory of the P^-bundles Y (k + 1; Ai , . . . , A/t) over Pj, were developed in Brieskorn's work [8] in the six­ties. (For another, purely geometric approach to the theory of rational scrolls over F} see Reid [3])

al i in i (4 fo it ain

P r p o s i t i o n 4.8 i h m o g e n e o u s binomial representation) ric va

riet Y ( r ; \ is embeddable int the projective space ' in fact

Z 0 , Z 2 2 « + 2 k Z l , k '

dente bihomogeneus coordinates it is representable as the zeroset

+ „ + for all triples (ß, ) with "• " "• 1 < » < * 1</* , + ^ "

It is possible to embed these, sometimes called Hirzebruch-Kleinschmidt varieties y (r; A Aft) into a single projective space by using the Segreembedding

l ) x p 4 l ) - l

but as it was proved by Ewald and Schmeinck in [19], this can be done in a more economical way (w.r.t. the degrees of the defining homogeneous binomi­als), namely by only considering quadrics within a suitably higher dimensional projective space.

Theorem 4.9 (Representation b quadrics).

The toric vaeties Y (r; A&) re embeddable int the projective space of diensi

—

depending n A,' an thei definin ideals (w.r.t homogenus dinates zo Z} of P generated by the quadratic binomia

0 v^, for all , ^ 3 ) € ( ,})

(c) The intersection theory on the varieties Y (r; A i , . . . , A&) is more complicated compared with that on F\ 's . For example, although for s = 1, the Chern numbers of the total spaces of these P*-bundles over Pj, for fixed dimension r = k + 1 are constant (cf. Brieskorn [8] Satz 24(i i) , p. 348), this is no more true for decom­posable P^-bundles over P£ with s > 2, because there is an obvious dependence on Aj's, i e . on the given "twisting numbers". On the other hand the isomorphism for the Picard group

Pic (y (r; A*)) Pic (P£) x x

is in general valid (cf. [26], Ex. 7.9, p. 170), and is useful as long as one makes a specific choice of a Z-basis and expresses each member of any examined r-tuple of divisors as concrete Z-linear combinations of its two elements. (The most natural

choice is to consider the classes in Pic corresponding to a typical fier of th bundl map and to a hyperplane section under the embedding of Y (r; A i , . . . , A&) in F^ of thm. 4.9, respectively). Let us now give, in broad outline, three complementary practical methods for the computation of intersection numbers

• First metho. For arbitrary and , there exist two towers of birational morphisms

Yo _!

r ( r ; _i

which are nothing but usual toric blow-ups with toric subvarieties of codimension > 2 as centers. (For the precise description of this procedure in terms of con­vex geometry, i.e. via barycentric stellar subdivisions of cones of A(r;\1,\2,—,\k) and the algorithmic determination of u, \i > 0, the reader is referred to Klein­schmidt [42], pp. 264-265). So the evaluation of intersection numbers of divisors on y (r; A i , . . . , Aft) can be reduced to another one w.r.t. divisors sitting on P£. The problem here is that one blows up and down subvarieties of varying dimen­sions and must therefore control carefully the intersection behaviour of the proper transforms of divisors in each step. The simplest example is Y (r; 1) (with k = 1, r = s + 1) which is P£ blown up at a (Tjv(r;1)-fixed) point (with v = 0 ß = 1); but for instance, already Y (3; 2) is the blow-up of P^ along an entire (TN3.2y fixed) curve followed by the contraction of another (3.üxed) curve (ie \i =

in this case)

• Second method. This method can be applied to any projective toric variety (or even to any compact toric variety), but demands familiarity with mixed volumes of "virtual" lattice polytopes. (Virtual polytopes are defined to be finite families of suitably translated dual cones of cones of a given fan, which are equipped with a Z-module structure w.r.t. formal addition and scalar multiplication, though their intersection might be not a polytope in the usual sence). If all the line bundles, being associated to the Cartier divisors whose intersection number is to be computed, are generated by their global sections (in other words, if the corresponding integral support functions are upper convex, cf. thm. 2.11), then we may apply formula (2.6) and evaluate the normalized mixed volumes of the arising lattice polytopes w.r.t. the dual lattice. For arbitrary Cartier divisors however, we need the combinatorial version of "moving lemma", i.e., to write down each of their associated support functions as the difference of two upper convex (or even strictly upper convex) support functions and to calculate afterwards the

desired intersecti numbers as the mixed volume of e difference of two virtual polytopes. (SeeinEwald'sbook[18],thm. V. 5.15, pp. 175-177, and thm. VII. 6.3 pp. 292-295. For an intrinsic, algorithmic characterization of moving lemma of this kind, we refer to Wessels' thesis [80], Satz 3.2.4, p. 83, and 3.2.18, p. 89 In contrast to [18], he works directly with "generalized" mixed volumes of virtual polytopes the normalizations of which take integer, but not necessarily only non-negative values). A simple example: the self-intersection number of the canonical divisor of Y (3; 2). Since Y (3; 2) is a Fano 3fold, —KY(3-2) is ample, and it is easy to verify that the lattice 3-polytope P-KY(S.2) (3-polytope, in the usual sence) induced by the anticanonical integral strictly upper convex support function, can be realized (up to an affine integral transformation) by

AV = C

i.e., by the polar of a lattice b y p i r i d o a t r iag le ialentl by a triangular lattice prism. Fig. 3 shows P-K)-An immediate calculation gives

K Y - 3; 3;2)

(b formula (25))

• Third method. If £ is a locally free sheaf with q = rk(£) defined over a smooth projective complex variety Z and

TT : F {£) = roj (Sym {£)) Z

the associated projective bundle, then by Grothendieck's direct construction of the Chern classes of £,

(£)€#&) q},

i e by setting c {£) = and

( £ ) £ « (4

(cf. Hartshorne [26], p. 29, or Fulton [21], 3.2.4, p. 55), the C o w ring A (F (£)) can be viewed as a free A (Z)module generated by the classes [ ] , [E\ - 1 ]

() 0 1 1

() 4 1

-1

() 1 1 1

() 1 1

-1

() 1

-4 -1

ig.

where here E denotes the divisor on V(£) corresponding to 0p(£) (1). The rela­tionship between the Chern polynomial of the tangent bundle of P (£) and tha t of the pullback of the tangent bundle of Z follows from the relative tangent bundle exact sequence (see Fulton [21], 3.2.11, p . 59). In part icular computing the first Chern class we deduce the anonical bunl formula :

Kv (K det (£)) - qE (43) lin

The equation (4.2) for Z = P£ and £ = OP« © 0 P « (Ai) © • • • © Op. (Xk) tu rns out to be a quite powerful tool for manipulat ing intersection numbers. In the next proposition we compute two basic self-intersection numbers for Y ( s + 1;A) (as P^-bundle over P£,) which will be used in § 8, and leave to the reader as exercise to examine further (and more general) examples of various r t u p l e s of divisors by applying the above mentioned methods

P r o p o s i t i o n 4 .10 . For k = 1, r = + 1, i = A ^ 0 the self-intersection number of the canonical divisor of the tta space of the Vjund Y ( ; A) — is given by the form

w £ (i) 2)" ryr__1 t44) = o ^ '

Moreover, the self-intersection number of the divisor E V (pos ( } ) ) (in the tati of 4 5 ) equa

A1-"1 (45)

Proof. For Z = Tsc, £ = 0 © O A) and H a divisor of P (£) associated to the

pullback *Ci (£) we have

E (E XH) = E A (H E) = (4

(by (42) {£) = ) and

=0, { r ~ l E ) l (47)

(by definition). Furthermore by the canonical bundle formula (43)

Y(r;X) det (£)) 2E ) H XH 2E = ( r) H 2E lin lin

For the selfintersection number we get

r, = ((Xr)H 2Ef £ 2 ) r " r) ^ (48) i=0

(by the first of equations (47)) and applying successively ( 4 ) and the second of the equations (47)

7- (XH) r--1 (H (XH) r~-2 • •

XH)^-1 r - _ 1 f-1 E) = r - _ 1

Thus, formula (4.4 follo i r l fr (4 i a l l (45 is i i l a l via ( 4 ) and (47)

5. Toric description of abelian quotient singularities

Abelian quotient singularities can be investigated by means of the theory of toric varieties in a direct manner. If G is a finite subgroup of GL(r C) then (C jG is automatically an algebraic torus embedded in Cr jG.

Notation. We shall henceforth use the following extra notation. For gN, ß € Z, we denote by [ß] the (uniquely determined) integer for which

0 < [ß]v < ß /^„(mod )

If q € Q, we define [q\ to be the greatest integer number < q. "gcd" will be ab­breviation for greatest common divisor, and diag(äi, . . . ,3 r) for complex diagonal

x r matrices having 3 , 3 r as diagonal elements. Furthermore, for integers v > 2, we denote by C,v : * the "first" ^ t h primitive root of unity.

(a) Let G be a finite subgroup of GL(r, C) which is small, i.e. with no pseudore flections, acting linearly on C , and p : C —> C jG the quotient map. Denote by ( C /G, [0]) the (germ of the) corresponding quotient singularity with [0] := p (0)

Proposition 5.1 ( i n g u l a r locus) IfG is a s l l finite subgroup of GL(r C) then

S i n g ( C 7 G ) = p ( { z e C | G z ^ { I d } } )

whe Gz : g € G \ g is the istropy group of = ( r) G Cr

Theorem 5.2 (Prill's group-theoretic isomorphism criterion). LetGi,G-2 be t ll finite subgroups of GL(r C). Then there exists an analytic isomor phis

( C 7 G [ 0 ] ) ( C 7 G 2 , [ 0 ] )

and ol if G and Gi ar onjugate t each ther within GL(r, C)

Proof. See Prill [57] thm. 2 p. 382, and Brieskorn [9], Satz 2.3 p. 34 •

(b) Let G be a finite, small, abelian subgroup of GL(r C), r > 2 having order G| > 2. Define

{ = ( , . . . , 0 , e = ( . ,

to denote the standard basis of Z r = Y t n e standard lattice its dual and

= Max-Spec [?f\ ]) = (C

Clearly

TNG = Max-Spec = ( C /G

is an r - d i m i o n a l algebraic torus with 1-characters . Using the exponential map

( J 0 RT = e x P ( y ) : ( e

/ = T K / = T V

and the injection t : M GL(r C) defined by

T T = t ( ) : = d i a « ( ) € G L ( r Q ,

we have obviousl

= (to exp)~ (G) ( and determinant det (NG) = — )

(as long as we choose eigencoordinates to diagonalize the action of the elements of G o n C ) with

If we define

Jm = Ji ' ' ' Jr is a Gnvar ia Laurent monomial (m = ( )) (an et )

= p

i m i iti fa

together ith its faces}

NN0 -

induced by the canonical duality pairing

M0/ /Vo ->• Q/Z ^ C

(cf. [22] p. 34 and [53] pp. 2223) we get as projection map :

X ( N 0 A ^ X ( N ) ,

where

X (N ) = =CT/G = Max-Spec ( c [ «-

• Formally, idntify [0] with orb(ff0)- Moreover, in these ter i n l a locus of X (N A ) can be written (by 5 and 25) as the union

*,<*<*»= O*M ( ^ w | ;i:ulf(:;(;^2})

• In particular, if the acting group G is cyclic, then, fixing diagonalization of the action on C we may assume that G is generated by the element

d iag (c r , c r for r integers a . . , a r e {0,1, ...,l — 1}, at least 2 of which are ^ 0. This r-tuple ( « i , . . . , ar) of weights is unique only up to the usual conjugacy relations (see 5 5 below) and is to be identified with the socalled lattice of weights

Q ( a a

containing all lattice points representing the elements of

G = { d i a g ( c r , . . I A e Z , 0 < A < } .

Definition 5.3. Under these conditions, we say that the quotient singularity X (N ) orb (a)) is of type

( a a 5

(This is identical with the definition given in [60], § 4.2, p. 370, up to the prede-terminated fixing of the primitive root C; of unity to be the "first" one. In fact this extra assumption is not a significant restriction, and by fixing in advance the isomorphism G NG/N0 one just simplifies certain technical arguments. Even if we would let (t denote an arbitrary primitive root of unity, all results would remain the same up to an obvious multiplication of the exponents of the diagonal ized elements of G by a suitable integer which would be relatively prime to ; see also the comments in [39] at the top of p. 225)

• The existence of torus-equivariant resolutions of cyclic quotient singularities was proved by Ehlers [17], § 1.3 & III.1-3, along the same lines as the more general the orem 3.9, i.e., by appropriate subdivisions of ero into smaller cones of multiplicity 1. (Essentially the same result expressed in the past language of gluings of affine pieces, is due to Fujiki [20], § 1.3).

• Note that, since G is small, gcd(, a.\_,..., cti a) = for alH r (The symbol means here that a is omitted

L 5.4. (i) lic qutien singurity of type (5. has splittin en si 2 r and l the exists an indexsuset

r-x r } ,

such that C*^l — Ctf2 ' 0 L V ^

which is in addition maximal w.r this prope

(ii) lic qutient scsingrity of ype (5 is isolated i and l

gcd (cci I) = 1, V 1 i r .

Proof. It is immediate by the way we let G act on C . o

(c) For two integers r >2 we define

A(ir) = («„«) , W g 1 i S t f , r and for ((c a) (a a)) e A ( ; r ) A (; r) the relation

(a a) ^ (c a

there exists a permutation 0 : { l , . . . , r } - > { , r } and an integer A, 1 < A < I — with gcd (A, ) = such that

It is easy to see that ^ is an eqivalence relati n A ; r ; r)

Corollar 5.5 (Isomorphism criterion for cyclic acting groups).

Let G, G' be two small, cylic unite subgroups of GL(r, C) acting on C and let the correspondin quotient singularities be of ype j ( c t i , . . , ar and jr (a[ a respectively Then the exists an analtic (torusequivariant isomorphis

X(NA)orb(a0)) (X(NA),orb(a0))

and only I and ( « . , a) ^ ([, a) within A ( ; r)

Proof. It follows easily from 52 (cf. Fujiki [ ] lemma 2 p. 2 9 )

Proposition 5.6 (Gorenstein-condition).

Let ( C / G , [0]) = (X (NG AG) ,orb (CT)) e an alian qutient sinrity. Then the followin conditions are equivalent :

(i) X (NG, AG) = Ua = Cr/G is Gorenstein

(ii)G SL(r,Q

(iii) { ( ) n) > for ll n, n f~l (N })

(iv) (NQ, AQ) , orb (a0) is canonical sinarity index

In particuar if(Cr/G, [0]) is clic of type j (« a) then (i)(iv) equiv ent t

= 0 (mod i

Proof. See Watanabe [78] and eid [58] . •

• If X (NQ, A Q ) is Gorenstein, then the cone a = pos ( ) is supported by the so-called junior lattice simple

= c

(w. NG, cf. [39], [4]). Note that up to 0 there is no ot lattice point o CTO H NG lying "under" the affine hyperplane of W containing SG- Moreover, the lattice points representing the / — 1 non-trivial group elements are exactly those belonging to the intersection of a dilation of with ar(eo) , for some integer A, 1 <

. Lattice triangulations and crepant projective resolutions f Gorenstein abelian quotient singularities

In this section we briefly formulate (and only partially prove) some general theo­rems concerning the study of projective, crepant resolutions of Gorenstein abelian quotient singularities in terms of appropriate lattice triangulations of the junior simplex. (For detailed expositions we refer to [12], [13]). Diagrams and formulae in boxes outline actually the only essential prerequisites for the reading of the rest of the paper.

(a) By vert(<S) we denote the set of vertices of a polyhedral complex S. By a trian-gulation T of a polyhedral complex S we mean a geometric simplicial subdivision of <S with vert(<S) C vert(T). A polytope P will be, as usual, identified with the polyhedral complex consisting of P itself together with all its faces

(b) A triangulation T of an r-dimensional polyhedral complex S is called coherent (or regular) if there exists a strictly upper convex T-support function ip : \T\ —> M, i.e. a piecewiselinear real function defined on the underlying space \T\ of T, for which

^ ( + ( l - ) y) > 'tp (x + ( rj) (y) , for all x, y g \T\, and € [ ] ,

so that for every maximal simplex of T, there is a linear function rjs : |s| —»-K satisfying tp (x) rjs (x), for all x e \T\, with equality being valid only for those x belonging to s. The set of all strictly upper convex T-support functions will be denoted by SUCSFR (T). A useful lemma to create a new "global" strictly upper convex support function by gluing together given "local" ones is the following :

L e m a 6.1 (Patching Lemma) t P W e a r l T = \ i (with I a unite set a coherent trianlati of P, and % sij \ j £ } (J nite, for all G I) a coherent riangulation of s», for all i £ I. If \ — dente strict uppe onvex suppor functions such that

fo all then

all the simplices i \ Mj j £ , and , £ / }

forms a coherent triangulation of the initial olyope because the above f)t can

e canonically "patched ogether" onstruct an eent of SUCSF I T )

Proof. See Bruns-GubeladgeTrung [ ] lemma 2 2 2 pp. 4 3 4 5 . •

(c) Let N denote an rdimensional lattice. By a lattice polytope (w.r.t. N) is meant a polytope in JVR= Rr with vertices belonging to N. If {no, n i , . . . , n^} is a set of k < r affinely independent lattice points the lattice fc-dimensional simplex

= c o n v ( { n o , r i 2 , . . . ,nu}), and Ns = lin( o 71* o}) f N, then

we say that is an elementary simplex if

y £ } n 0, n 0 nk 0} .

is basic if it has anyone of the following equivalent properties:

(i) no r o • •, n* o} is a Z-basis of

(ii) has relative volume Vol( ) = -———• — (w.rt ) . det (Ns)

Lemm 6.2. (i) Every basic lattice smplex is eentary (ii) lementary lattice silices of diension < ar asic

Proof, (i) Let = c o n v ( o n n&}) be a basic lattice simplex. Since

i - 0 n 0 . , n k

is a Z-basis of iVs, if n fl (A }) then obviously n for some index Thus

y £ } n 0, m 0 nk 0} .

(ii) The relative volume of an elementary lattice 1-simlex (resp of an elementary lattice 2simplex) is always equal to (resp. equal to /2)

mple 3

= c ( { 0 , e i , e 2 , . . , e r _ 2 , e r , _ i , ( , , . . . , , l , ) } ) C l , r > 3,

(w.r.t. Z r) serves as example of an elementary but non-basic simplex because n Zr = ver t ( ) and

r! Vol (s; Z ) = |det ( e u , e r_i ( , . . . , 1 , 2 ) T ) | = 2

Definition 6.4. A triangulation T of a lattice polytope P C ./VR = W (w.r.t N) is called lattice triangulation if vert(P) C vert(T) C N. The set of all lattice triangulations of a lattice polytope (w.r.t N) will be denoted by LTRjy )

Definition 6.5. A lattice triangulation T o f F c i V R = Mr (w.r.t. N) is called maximal triangulation if vert(T) = N n P. A lattice triangulation T of P is obviously maximal if and only if each simplex s of T is elementary. A lattice triangulation T of is said to be basic if T consists of exclusively basic simplices We define :

) : T £ ) | 7" is a maximal triangulation of P } ,

asi ( j . = | r G p ^ | 7 is a basic triangulation of P } .

(Moreover, adding the prefix Coh- to anyone of the above sets we shall mean the subsets of their elements which are coherent). The hierarchy of lattice triangula­tions of a (as above) is given by the following inclusion-diagram:

si

o h s i oh oh

roposition 6.6. For y lttice pol NR W (w th set maxi coherent rianlations oh Coh of is nn-empty

Proof. Consider a s c p . cone supported by i W1 and then make use of [54] cor. 3 8 p. 394 o

R e m a r 6.7. (i) Already in dimension r = 2 there are lots of examples of with

L T Rbasi ) x o h s i ) ^

See, for instance, 7.5 below.

(ii) In addition, for lattice polytopes ^ ) ^ 0 does not imply neces saril

CohLTR^as ic (P) ^ 0 .

As it was proved recently by Takayuki Hibi, there exists a 9-dimensional 0/ polytope (with 5 vertices) possessing basic triangulations but no coherent an basic ones

(d) To pass from t i a l a i i n l a i i in extra notation

Definition 6.8. Let (X (NG, A G ) , orb(eo)) be an r-dimensional abelian Goren-stein quotient singularity (r > 2), and sG the (r )-dimensional junior simplex For any simplex of a lattice triangulation T of sG let as denote the s c p . cone

{Ay e ( J G ) R | A € M>0 , y £ } ( = pos ( ) within (N

supporting . We define the fan

T):

of s c p . cones in (i W

(X(N

(X(A

(i

(Whenever we put the prefix QP- in the front of anyone of them, we shall mean the corresponding subsets of them consisting of those desingularizations whose overlying spaces are quasiprojective

Theorem 6.9 (Desingularizing by triangulations)

Let (X (NG, AG) ,orb(<7o)) be an r-dimensional abelian Gorenstein qutient sin rity (r > 2). Tien the exist oneo-one orrespondences

R ^ i l: X ( i A ) )

( X ( J ) )

( X ( J ) )

partial crepani T/vG-euivaria desingularizations of X (NG, A

with overlying spaces having at most (Q-factorial) canonical

singularities (of index

partial crepant TjyG-equivariant desingularizations of X (NG, A

with overlying spaces having at most (Q-factorial) terminal

singularities (of index )

crepant T/vG-equivariant (full) desingularizations of X ( i A

o h ^ l:

X ( i ) )

o h ^ (X (N ))

oh ( X ( J A ) )

which ealized by an T/vG equivarian rationa orphis of the form

induced by appin

T ^ T ) (T) . • ( (T)

Sketch of proof. X (iVG, AG) is Gorenstein and has at most rational singularities ie . canonical singularities of index . Moreover its dualizing sheaf is trivial. Let

/ = id : X A X (N A

denote an arbitrary partial desingularization Studying either the behaviour of the

highest rational differentials on X (iVG, AG J (see [58], § 3, [ ] , § 4 8 6], prop.

3, or [12], prop. 4 ) or the support function associated to K N ^ (cf. [4]

§ 2) one proves

N G A = f ( < ( ) ™ ( < ? ) ) ) A S(i

where D V (Q) = V (pos (n {Q}}))- Obviously / is crepant if and only if

Gen ( < y = ( € (i

and since the number of crepant exceptional prime divisors is independent of the specific choice of / , the first and second 1-1 correspondences (from below) of the first box are obvious by the adjunction-theoretic definition of terminal (resp

canonical) singulaiti In particular, all TNG-equivariant partial crepant desingu larizations of X (NG, AG) of the form (6.1) have overlying spaces with at most Q-factorial singularities, and conversely, each partial TNG-equivariant crepant desin-gularization with overlying space admitting at most Q-factorial singularities, has to be of this form. (Q-factoriality is here equivalent to the consideration only of triangulations instead of more general polyhedral subdivisions. Furthermore, by maximal triangulations you exhaust all the crepant exceptional prime divisors) The top correspondence of the first box follows from the equivalence

mult (a; NG) = for the cone

(T) supporting it

It remains to prove the 1-1 correspondences of the second box. As it was explained

in [12], § 4, for every i> e SUCSFQ (NG, A G (T)) , the restriction ip \j- belongs to

S U C S F R (T), and conversely, to any support function ip g S U C S F R (T), one may canonically assign a rational (or even an integral) strictly upper convex support

function defined on i

apply corollary 2 3 (T ish t

Remark 6.10. (i) Flops. Every pair of triangulations 7 7vG

gives rise to the determination of a birational morphism

x ( ( T ) x (

which is composed of a finite number of flops (cf. [54], § 3). Since all triangulations of C o h - L T v G (sG) are parametrized by the vertices of the so-called secondary polytope of sG, this transition-map is induced by performing successively bistel lar operations, i.e., by passing from the vertex of the secondary polytope of sG

representing 71 to that one representing T2 following a (not necessarily uniquely determined) path which connects these two vertices. (In dimension 3 this is noth­ing but Danilov's theorem [14], cf. 7.4 (vi) below). For detailed presentations of the theory of secondary polytopes we refer to the articles of Billera-Filliman-Sturmfels [7] and OdaPark [54] and to the treatment of Gelfand-Kapranov-Zelevinsky [23] ch. 7.

(ii) Factorization. The birational morphisms corresponding to members of QP-PCDE (X (NG, A G ) ) can be decomposed into more elementary toric contrac tions by Reid's toric version of "MMP" ([59], (0.2)-(0.3)). In several cases these contractions can be directly interpreted as inversed (normalized) blow-ups. For concrete examples see rem. 7.2 and § 9 below.

(iii) Exceptional divisors. Each irreducible component of an exceptional divisor w.r.t. an / = fj (as above) is a Q-Cartier prime divisor which carries itself the structure of an (r — l)-dimensional toric variety determined by the corresponding star withi A G 7") (cf. above § 2 , (ii)). An exceptional prime divisor is

com if l if l a i c i in it l in(c (s

2 )

Theorem 6.11 (Number-theoretic version of K y correspondence).

Let (X (i A ) orb(co)) be an lian Gorenstein qutient sinrit of di > 2

f = id : X X (N

a TV G equivariant crepant full resoluti and : = / 1 ([0]) the cent ve the orgin 0] = orb(e

6A^ | n(ß)G int ( a 0 ) n

V (°) o e Z\ ) , int (a) int (a0)

Then F is a stron deformatin retract of X NQ, A G , and on the even cho-

mology groups of rivial. Thei diensions (over Q) given by the formlae

In paticuar the tologica er-Poincare characteristic ofX ( equa

3)

if

d i F 2 Q ) = ( ( a r ( ( ) n ) if

otherwise

( x ) ) ) = G|

bviusly, the numbes ( 2 ) (63) are independent of the choice of trianlations

of by means of which we constuct 7")) (cf. ( ) )

Proof. See Batyrev-Dais [4] thm. 54 p. and [13] for further comments

Corollary 6.12. If ( C / G , [0]) = (X (JVG AG) ,orb(a0)) is a Gorenstein cyclic quotient singarity of type j (O a) then maintainin the above tati and assumpti w obtain

Remark 6.13. The numbers of the right h side of (6.2) and (6.4) make sence even without assuming the existence of a TG-equivariant crepant, full desingu-larization of X (NQ,AQ), and were used in [4] as the "correctional terms" for introducing the formal definition of the so-called string-theoretic Hodge numbers of Calabi-Yau varieties (or, more general, of Gorenstein compact complex varieties) which have at most abelian quotient singularities

(e) By theorem 6.9 it is now clear that Reid's question (formulated in 1), re­stricted to the category of torusequivariant desingularizations of X (NA) can be restated as follows :

uestion For Gorenstein abelia quotient singularit

( X ( i A ) o r b ( o ) )

with junior simplex SQ , what ind of conditions on the acting group would guarantee the e i s t e n c e f a basic, coherent triangulation s<j

Though (as already mentioned in §1) this question will be treated explicitly in [13], we at least shall explain here how a very simple necessary existencecriterion works and apply it efficiently in our special singularity-series i § 8.

emm 6.14. Let (X (NGA) o r b ( e ) ) be a Gorenstein alian quotient sin gularity and the junior simplex. If BQ adits a asic trianlati T, then for

ll n £ {N }) nC the exist r lattice points n elogin n N that

n £ >0 ii + Z>0 r

Proof. Since T is a basic triangulation inducing a subdivision of o into cones of multiplicity 1, n belongs to a subcone er' of co of the form

= M>o ni + • • + M>o nr, mult (a ) =

and can be therefore written linear combination

n = / ^n i 1 (fit £ M>o, V«, 1 i < r)

Now r} is a Z-basis of which means that n can be also written as

n = ßnx + (/ € Z, r)

By linear independence we get ß £ Z IR>o = Z>0 r

Theorem 6.15 (Necessar ExistenceCriterion)

Let (X (NG,AG) ,ovb(ao)) be a Gorenstein lian qutient sinrity If admits a asic trianlati T, then

lhNG (a0) 5)

.e. ll of th H sis ofa eith ents o b g . , e}

Proof. The inclusion "D" is always true (without any further assumption about the existence or non-existence of such a triangulation) and is obvious by the definition of Hilbert basis. Now if there were an element n G Hlt>7vG (co) \ (SG H NQ), then by lemma 4 this could be written as a non-negative integer linear combination

n = ßx • •

of r elements of s G H NQ. Since 0 ^ lb7vG (00), there were at least one index j = j , G { r } , for which ß -# ^ . If jm = 1 and ßj = 0 for all j j € { 1 , . . . , r} \ { j , } , then n = n j , £ sG n iVG which would contradict to our assumption. But even the cases in which either ß -e = 1 and some other /x - s were 7 , or ß jm > 2, would be exluded as impossible because of the characterization (2 . ) of the Hilbert basis Hlb7v (co) a s the set of additively irreducible vectors of C n (N }). Hence HlbivG (C 0 ) C NG.

Remark 6.16. (i) For a long time it was expected that condition (6.5) might be sufficient too for the existence of a basic triangulation of SG, and, as we shall see in [13], this is the case for some special choices of cyclic group actions on C . Never theless, Robert Firla (a student of Günter M. Ziegler) discovered recently the first counterexamples. Among them, the counterexample of the 4-dimensional Goren-stein cyclic quotient singularity with the smallest possible acting group-order, ful filling property ( 5 ) and admitting no crepant, torusequivariant resolutions, is that of type ^ (1,5,8,25).

(ii) To apply necessary criterion 6.15 in practice, in order to exclude "candidates" for having crepant, torus-equivariant resolutions, one has first to determine all the elements of the Hilbert basis HlbjvG (CTO) and then to test if at least one of them breaks away from the junior simplex or not. From the point of view of complexity theory of algorithms however this procedure might be "NP-hard" (cf Henk-Weismantel 27], § 3)

Exercise 6.17. For the singularity of type | (1,2,3,3) determine explicitly the Hilbert basis HlbjyG (co) and show that it does not possess any crepant torus equivariant resolution because condition 5) is violated Hint. Verify that ±5 £mhNa(o)}

7. Pecliari t ies of dimensions 2 and 3

In the low dimensions r e {2,3}, we have always

E S ( X ( i V G A G ) ) = C D E S m a x ( X ( J A G ) )

by lemma 6.2 (ii) and thm. 6.9. This is exactly the lemma which fails in general if r > 4 and makes the high dimensions so exciting cf. 6 . . However low

dimensional Gorensein abelian quotient singularities are still valuable as testing ground for lots of interesting related prolems and pave the way to systematic generalizations.

(a) In dimension 2 we meet only the "classical" A\\singularities e. cyclic quotient singularities of type l/l{l,l — 1)).

Lemma 7 .1. All 2-dimensional Gorenstein abelian quotient singularities

(X(iVA G ) ,orb(<7o))

are cyclic of type j (1,1 — 1 with / > 2). They admit a unique projective crepant minimal resolution 6.1) induced by the triangulation

T = { conv ( l (j -1,1-(j- 1))T \ (jl - ) T } ) | 1 < < ?

and the I — 1 exceptional prime divisors

Dj := V (pos ( j (jl - )J } ) ) , 1 < < I - 1,

are smth ratio ves in intersetio ers

A • Dj) = <

, if \ =

, if i=

, if > 2

fo all

Remark 7.2. Obviously, the subdivision of c into / cones of multiplicity 1 in duced by the above T can be done successively in k "steps", with 1 < k < I — 1, i.e., by drawing in k steps the / — 1 required rays in any order you would wish. (The possibility of drawing more than one rays in a step is not excluded). Such a procedure gives rise to decomposing the full desingularization into a series of partial ones. It should nevertheless be stressed that starting-points of different se ries of choices correspond to blow-ups of different ideal sheaves I with orb(ao) supp(C[/CT /I). (Note that in dimension 2 we do not need extra normalizations

onsequently, there are lots of factorizations of the crepant resolution morphism = ff. Let us illustrate it by considering the example of the ^s ingula r i ty

i 1,4 The morphism admits of two different natural factorizations

= 92 ° 9i = h o h3 o h2 o h

which are depicted in figures 4 and 5 respectively.

Since

K ) = {mi ( 5 , m2 = 1 , 1 , m ( 0 ) }

using the induced e m d d i n g i: U •—> C of X (N A )

= {(z C3 I 0 } , Vi, 1 < 3,

and proposition 3.8, we o t a i n i and <?2 coming from lowing up only maximal dimensional ideals:

X ( , A T ) ) = B ) )) ^ ) -21

More precisely we first sudivide <J into the three sucones 2, 3, with

{y € ITO I ( - m y ) > 0 & - m y > > o s | f e

{y G 0o | ( - m y ) > 0 & m 3 - m y ) > os ((§ ± ± f

{y € 0o | ( - m y ) > - m y ) > po f f

fter that we perform again a usual blow-up, but now for the cone instead of which contains the remaining two lattice points.

1(1,4)T

-15(2,3)T

v T(3,2)T

s 0 \15(4,1)T= ( )

ig

(ii) In the second factorization i gives rise to a "starring" sudivision of into only two cones

In fact

ID 2}) and o s } £ }

is a directed low-up w.r.t. ( l / 5 , 4 / ) T s öu„ ideal sheaf I can be taken the pullack via of the ideal sheaf J = J~ supporting the axis of with

{z in <£[z-i\), ecause

y S - m i , y ) > and y ) >

y e mi ~m2y) > and mi + m - m y ) >

Another characterization of Blorb((Toi (JT^) avoiding the embedding is provided by the following commutative diagram due to the extension of the group action o n d ( )

) - ^ ) / G

C /G

Clearly, our blown up space is isomorphic to the quotien of the blown up C at the origin divided by G. For h2 we proceed analogously by letting a2 play the role of (To- Oviously the same is valid for / j 3 and /14. (Comparing the a o v e two factorizations we see that the first one is "doule speedy".

55

h4

e 1!

ig

iii) Finally, let us oint out that also itself can similarly regarded as a low-up morphism

/ ^ U<

s X one may choose the pullack of with the ideal

in z3

ecaus

and

m m i y ) > 0 3m2 - 2 m , y ) > 0

m + m3 - 2 m y > >

2 m 3 - 2 m y > >

mi - m 2 , y ) > 2m2 - m i , y ) >

m - m i , y > > ,

2m - mi - my) >

o s ü

o s ( | ! ( I 4 )

y G

y G

and finall

y G

2mi - 3 2 , y ) >

mi - 2 m , y ) >

m - 2 m y ) >

2m 3m 2 ,y ) >

2 mi - m2 - m y ) > m - m y ) >

2m2 - m 3 , y ) >

- m y ) >

2 m i - 2 m 3 , y ) > 0 ,

mi + m 2 - 2 m y ) >

3m2 - 2m 3 ,y ) >

- m y ) >

o s i i i i

o s ( ! ! ( ! | )

m} Exercise 7.3. To represen / as the restriction (over Uao) of a proper birational morphism which comes from a single blow-up of a torus-invariant 0-dimensional ideal of C[^i,2;2,23] supporting only 0 £ o C cf. thm. 3 .1) it is enough instead of the the a o v e to consider

zl ) .

(From the point of view of toric geometry, this means that one has t find among the defining monomials suitable monomials involving powers of the variables Z\ and Z3 separately such that the corresponding order function comes again linear precisel on the a o v e five maximal cones.

(b) We now focus our attention to the new phenomena which arise in dimension three.

Theorem 7.4 ( W a t happen in the "intermediate" dimension 3 ?).

Let (C3/G, [0]) = (X (NQ, A Q ) ,orb (<T0)) a3-dimensional Gorenstein abelian quo­tient msc-singularity and T a maximal (and therefore basic triangulation of the junior simplex S inducing a crepant full resolution

, A X (N A

For any n G vert( let Dn denote the closure V (pos ({n})).

(i) Ifn G int(sGr) n c then Dn is a rational compact surface coming from usual blow-ups either of P or of a Hirzebruch surface ¥\ at finitely many T J O S J I }

ßxed points

ii) If dsG H ( A r G \ , e 2 , e 3 } ) ^ 0 , and n G conv(e i le i 2) \ { ^ e ^ } , with {*i,*2} C {1,2,3}, i\ 7 %i, and {«3} = {1,2,3} \ «i,«2}, then Dn is the total space of a ruled Gbration over the "iaxis" of Its fbers over the punctured 1 a x i s are isomorphic to P j .

iii For three distinct vertices n,n'n" ofT, we have

n n n if conv {n, n' n"\) is a 2-simplex of T Dn-Dn,-Dn„ otherwise

(iv) If n,n' G vert (TO, conv{n,n'}) is a 1-simplex ofT, but no both n and n' belong the same face of OBQ, then there exist exactly two vertices t), t)' ofT, such that conv({n,n', t)}), conv({n,n' i)'}) are 2-simplices ofT satisfying a Z-linear dependency equation of the form

+ t + t , for some unique G Z with — 2 1

and K=(D2

n-Dn,), (Dn-D2n,)

Furthermore, the normal bundle of the rational intersection curve

= V p o { n , n ) )

splits into the direct sum:

^ C / X , A ) ®

IfnG i n t ) H then selfintersetio er

D = 12 - en Star po {n AG))) (7

(vi) For any other maximal triangulation T ofs, there exists a birational mor phism isomorphism in codimension 1

, A^ , A

which is a composite of finitely many elementary transformations simple flops w.rt. rational smooth curves

Proof, ( follows from da's classification of smooth compact toric surfaces ([53] thm. 1.28), (ii) is clear by construction and iii) by (2.3). The vectorial Z-linear dependency equation 1 of (iv) with + — 2 is obvious because n, n', t), t)1

are "junior" elements; 7.2) follows from 2. and (7.3) from the splitting principle of holomorphic vector undles over the normal undle exact sequence

and the triviality of the dualizing sheaf of Q, A G ( ) J ) in ( ) can

prove by making use of adjunction formula, combined with Noether's formula and x Dn, OD„) = 1- For the proof of vi we refer to Danilov 14 prop. 2 or Oda [3] , prop. 1.0 ( i i . •

Remark 7.5. In dimension three one has Q C D S (X (N A ) ) ^ cause

L ^ i ) = L R ^ x ) ?

by prop. 6.6 and thm. 6.9. Nevertheless, in contrast to what takes place in di mension 2, there exist lots of examples of finite abelian subgroups G of SL(3,C) acting linearl on C3 whose junior simplex SQ admits basic, non-coherent tri-angulations T which in turn induce crepant, full, non-projective desingulariza-tions of X (AT(3 A Q ) . simple example of this kind comes into being by taking G Z / 4 Z ) x (Z /4Z) to be defined as the abelian subroup of SL(3,C) gener ated by the diagonal elements diag(C4,C4> l) and diag( l , ( 4 , (4) , and T the tri angulation of figure 6. T suffers from the "whirlpool-syndrome" which makes the application of patching lemma 6.1 impossible, though strictly upper convex sup­port functions can be defined on each of its 2-simplices. The incompatibility of these local strictly upper convex support functions along the intersection loci of 1simplices can be also explained by means of their "heights" (cf. [72], p. 64); the assertion of the existence of a global upper convex support function on \T\ would lead to a system of a finite n u m e r of inconsistent integer linear inequalities and hence to a contradiction.

ig

P r o p o i o n 7.6 ( o m g y dimensions).

Let (C3 /G, [0]) = (X (No, AG) , or ((To)) denote a 3-dimensional Gorenstein cyclic quotient singularity of type l/l (01,012,013). Then the dimensions of the non-trivial

cohomology groups of the overlying spaces of any TNG -equivariant

crepant full desingularization of X (i are given by the formulae:

if

d i m 0 F ; Q \ ^ g c d ( a 0 if

\ Z - ^ g c d ^ O + 1 if

Prof. Since # (d (sG) fl iVG) = ^7=1 SC(i(oj, I) the lattice points representing the inverses of junior group elements elong to 2s and

G n VG) ( ( 2 s ) n ar (a0) n NG) I -

we have

int G) D ((2sG) n Par (<T0) H ) = 1 / - £ cd (o + 1

and the a o v e formulae follow from 6. 6.

E x r c i s e 7.7. Generalize prop. 7.6 for arbitrary abelian acting groups. [Hint Fix a splitting of into clic groups. Use denumerants of weighted partitions instead of cd's.

Proposition 7.8 (Uniqueness criterion in dimension 3). Up to isomorphism, the 3-dimensional Gorenstein abelian quotient msc-singularities (C3/G, [0]) = (X (N or ( I ) ) which admit a unique full resolution, are cyclic of type either

y 1 - 2 or

ü) \

In case ) there are [fj exceptional prime divisors

= V (pos ( { « } ) ) , « j (j, j , I - 1 < j I

all of whose are compact up to the last one for I even see fig. 7 and 8 particular one has

m if i < j i - i

if ~Y- I odd >

if \ I even

The "2>-dimensionaF [|J |J |J intersection-number-matrix is determined by

if ^ l - if i =

if i= if

i-DrDk if

if i = I - / 2 I odd

if i = I - / 2 «odd

if i= l/2 I even

if i = I / 2 I even

for all 1 < i < j < k < [|J (and by the usual symmetric property of intersection numbers). Moreover, the non-trivial cohomology dimensions of the desingularizing

space equal 1 |J and ^r\ respectivel by prop.

ii), ept na

Dn = V pos })) Dn,= V pos })) Dn„ = V pos ))

with

n i ( l 4 ) \ = i ( 2 , ) T „ = ± ( , 1 ) T ,

each of wiiicii is isomorphic to ¥2. They intersect each other paiwise along three rational curves which play interchangeably the roles of the fibers and of the 0 sections of the three projectivized P^-bundles ¥2 —> P^ (see fig. 9). Obviously, in both cases and ii the desingularizing birational morphism is projective by prop. 6.

Proof. The uniqueness (up to automorphisms of aff(sG) n NQ) of the triangulation T of SQ (inducing a unique crepant desingularization, up to isomorphism means that for ever 4-tuple of distinct elements of S

convnVnVnV

for all {^1,^,^3} C so that conv }) cannot a convex quadrilateral. Since

splcod or (c U„

it is eas to prove that cannot be abelian, noncyclic. For cyclic G"s acting on C by type 1// (« the uniqueness condition will e examined in two different cases.

• If the cardinal number of SQ H (NQ \ {ei, e2,63}) is > 4, then (7.5) is equivalent to say that all points of it lie on a straight line going through precisely one of the vertices ei,e2 of SQ (but of course # (dso H (NQ \ ei,e2,es})) G {0,1} because otherwise (7.6) would violated). This occurs onl in the case in which at least two of the weights , are equal ut then

3) ^ / — 2 within A /; 3))

(check it or see lemma 8.1 elow and therefore /G must have t p e of the form

( N ) ) then we get the inequalit

^ ^ c d ( a 2 <

which is valid onl for

, . 1/ , 1/4 , 1/5 ( , , [ a 3 ) \ 1/5 ( 1/6 1/ 1/7 ( )

62

(It s u c e s to assume +03 = ecause ( « 0 3 ) ^ (I — a.,l a-2,l 03) within A (I; 3)). Since ( 2 2 ) ^ ( , 3 ) within A (5 3), we see that from the above 7 possile types only (ii) 1/7 (1, 2,4) is inequivalent ( w . t . " ) to all those of the form (i The non-vertex lattice points of BQ for (C3/G, eing of this "new" ype (ii) satisfy viously ( 7 ) and the proof is completed.

• That the [|J exceptional prime divisors Dj have the structure given above is an immediate consequence of the more general thm. 8.4 of the next section which will be proved for all dimensions. The intersection numers • D • D are computale

Since ei 3 = 2 n 2 =

the above assertion for the structure of the exceptional prime divisors Dn, Dn* is obvious e.g. 4. For another proof see Roan-Yau pp. 2 2 - 2 3 .

Dl-1 = P2

(z 3-axis)

ig I odd

n( 12) = ( 1

2, 1

2,0) (z3 -axis)

ig I even

ig

64

8. On the m o n o p a m e i z d s i n g u l a i t y - i e

y ( l l / - ( r l ) )

This section contains our main results. Motivated 7.1, the uniqueness crite rion 7.8 (i) in dimension 3, and Reid's remark [62], .4, (concerning dimension 4) we study the monoparametrized singularityseries of arbitrary dimension with the simplest possible "latticegeometry, i.e. those Gorenstein cyclic quotient singu­larities whose junior simplex encloses onl lattice points l i n g on a single straight line.

Lemma 8.1. Let (C r /G, [0]) = (X (NG, G) orb(o-0)) be the Gorenstein cyclic

quotient msc-singularity of type — ( « , . , r) (with I = \G\ > r > 2) for which

at Jeast r — 1 of its defining weights are equal. Then in the notation of § (c)

« 1 . , / — (r — 1)) within I; r))

Proof. Using a permutation sending the r — 1 equal defining weights of the ove singularit to the first r — 1 positions we have

(a (ß,ß

for an integer ß, 1 < ß < I — 1. Since contains no pseudorefiections, gcd( = 1 i.e. there exists an integer 1 < I — with g c d ) = and A such that

+ \l (8.1

Moreover since /G is orenstein there must e an integer 1 < < r—1 such that

r - 1 8.

Equalities 8.1 and 8. impl

r - l \ - l l-r-l))

Hence

(ß / - r - 1 ) )

and we are done.

By the above lemma and cor we can obviousl restrict ourselves to the stud of singularities of t p e 8.

Theorem 2 ( R e s o i o n y a nique projective, crepant morphism).

Let (Cr /G, [0 (X ( Q) or (CTO)) be the Gorenstein cyclic quotient singu­larity of type

3.

with I = \G\ > r > 2 Then we have :

This msc-singularity is isolated if and only if gcd(/, r — 1) = 1.

ii Up to affine integral transformation, there exists a unique triangulation

inducing a unique isomorphism class of crepant Tequivariant partial resolution-morphism

(NGAG)

of X (NG,AQ) (with overlying space Q-factorial, and maximal with respect to non-discrepancy).

iii) T G C L R ^ x ie, f is projective

(iv) T € Coh-LTR^1 0 (SQ) (in other words, gives rise to a full projective, crepant desingularization) if and only if

i 1A)

e., iff either / = 0 mod( 1) or / = 1 mod(r — 1).)

(v) For I satisfying condition 8.4), the dimensions of the non-trivial cohomology

groups of the resolving space are given by the formulae :

8.

of, i) This follows directl from 4. ote that if gcd 1) > then

S i n g ) ) p o s _ 2 i ) )

e. the singular locus of X iVG C r / G is the entire " a x i s " of CT

ii) Let us first introduce some notation and make certain preparator remarks.

for i =

d i m ( i Q ) = bh for i e r - 2

l for i = r — 1

66

Define the v e o r

l )

if

if ? - i

and denote y n 1 < i < the th coordinate of each n I — 1 within =< W have

for some fixed j / — 1 [=i then obviousl

E Vf 1 < < k < I - 1

Since

E « y ( ( r - l ) . Z - r - l )

we o a i n the inclusion

} u | j )

nd conversely, for all 1 < < \_fzj\ we have [j (I — (r — l)) l — r — l)

which means that ^ 1- Thus we get the equalit

8.

• Construction of T• first define

& r - r - l ^ • • •

and

as well as

;? j 1 ] j \ e e e ? e ?

s ( ; £ ) : conv ( ; ^ ))

and

; £ ,...^ P° s ! £ ))

for all

and all r— possile r — 2) -tuples (£

All lattice points -^zj lie on a straight line of namel on

conv ( { ( L ^ J

and the can ordered canonicall

(L^x (L^r

via the increasing ordering of the enumeratorsuperscripts ) .

ext define the simplicial sudivision T having support ) as follows :

if mod r - 1))

all faces of ; . otherwise

COnv as

wher

and

as ( L ^

all faces of the simplices for all 1 < j ^ - and

( ; f all (^

• T is a triangulation. The proof of the assertion that the intersection of two arbitrary simplices of T is either a face of oth or the empt set is left as an eas exercise to the reader

• \T\ = SQ, ie . its support covers the entire junior simplex. It is straightforward to show that

«>nv ; & ) )

det e€ e? ei

hh - l ) - l - r - l ) ) - l - - l ) r - l ) \ = ^ 8. r! M

and

c o n v a s r\

Thus the a s e r i o n is an immediate consequence of the qualit

Vol conv , ( simplex of 7

Hi l) + Vol conv ))

• T is a maximal triangulation. This is ovious ecause sG ve r tT) the construction of T and (8.6) • T is unique. Suppose s is an a r i t r a ry elementary r — l)-simplex, such that vert(s) iVG learl s C cause itself is non-elementary. This implies

vert (s) ( G NG e i , . . ) ^ 0

Since n ^ s are collinear and s elementary simplex verts) contains at most 2 consecutive lattice points among them. Hence

v e r t s ) ) )

If this n u m e r equals 2, then s has to e of the form ( ; ^ , £ r ) , for some

[fzj \ and ( x , r ) ( y definition). Otherwise s possesses

only one vertex elonging to the relative interior of sG- However since the n^ are collinear and n^ = we must have I mod r — 1) and s as- Hence

is uniquel determined

(iii) The coherence of T will be proved by induction on the n u m e r jrriJ • The case in which this equals 1 is trivial. Suppose ^~fj > 1- It is eas to check that the simplex

?: conv ^ i \

can be mapped an affine integral transformation onto the junior simplex of an equidimensional orenstein ccl ic quotient singularit of t p e

l)

wit - ( r - l ) r - l

- 1 I

r - l - 1

I r - l

By induction hypothesis and by construction wejnay therefore assume that the restriction T of T on s is coherent hoose a S U and use the abbreviation

l ; l 2 , r , r - 2 ) l; 1 , t - 1 + 1 , . . . r - l ) l ; 2 r - r - l )

for i = Vt 2 < i < r - 2 for r — 1

69

ote that for each and <r 1 there exists an y' such that

l i n c o n v ) ) 1 - ) y for a

ecause of the convexit of OW define support functions S J

^ y ^ ^ y ) - i - ) +

One verifies easil that S U S t) and

t |snSl snS l

for all (-, 1 < i < r — 1. pp l ing patching lemma 6.1 we get S T) and we are done.

• An alternative (but not directly constructive) method for showing the coherence of T is to combine the uniqueness of T with prop. 6.6.

(iv) Since T is uniquely determined the volume formulae in (8.7) and 8.8) show that is a full, crepant torusequivariant resolution-morphism if and only if [I r _ i G {0,1}. Another more direct wa to verif that condition 8.4) is necessar is the following: If we assume that 1 the group order / can written as

ote that

l) + i with 1 - 2

CO l _ L±fH l<

1.

and that the decreasing sequence

(L^T

has minimum element

• • •

JL- HL-

8.1

We shall discuss the two possile cases w.r.t. the values taken gcd 1) separately.

I) gcd r — 1) 1. Then there exists a .,l — 1 such that

l - r - l ) mod / ) .

learl

« - 1 ) i J L - 8.11)

and consquent

K —— l - l

by 8.6) and 8 .1) . If there were a basic triangulation of s then n^ would not belong to HlbjvG ((To) y the equality (6.5) of thm. 6.15), but it would be representable as a linear combination of (at least two) elements of the Hilbert basis Hlbjv ((To) with positive integer coecients . This would oviousl contradict to 8.11) ecause of 8 .1) .

II) gcdr — 1) > 2. In this case define I / cd( r — 1) Then

l - r - l ) mod /

and repeating the same argumentation for this new as in I) we arrive again at a contradicting conclusion.

v) Formula 8 . ) follows from 6 . ) . In particular

dim0 , A # ^ + 1 < ^

and the proof of the theorem is completed

Remark 8.3. i) To prove the necessity of condition 8.4) for T to be basic we preferred to make use of "Hilbert-base technology ecause it is generally applicable to any Gorenstein abelian quotient singularity. Alternative ad hoc methods (for the above special situation) are either a sui ta le direct manipulation of determinants or the use of normalized-volume-bound. According to the latter one, violation of (8.4) would impl for the topological Euler-Poincare characteristic of the overlying space:

( G , A T ) ) ) T) r)

( ( / • - 1) -simplices of T) 2 . ) )

^ Sr)) 1 = ^ r - 1) + 1

l - l + l = \ 8 . ) )

which would impossile for asic T y 6 . ) .

ii) In fact, if [ / ] r _ 1 ^ {0,1} which is possile only for r > 4), the only non-asic r — l)simplex of T i s c o n v a s ) . . If gcdr — 1) 1 then the toric variet

N a T J has orenstein terminal isolated clic quotient singularit

of t p e

- 1 , < - - i ) , i

—l) /

lying on the a n e piece f7pos(BiaBt) • Otherwise, the singular locus is not a singleton more precisely, it is 1-dimensional, e., the corresponding quotient singularity has splitting codimension r — 1, and can e viewed as a 1parameter "Schar" (« crowd of terminal singularities of t p e

Z=l v l) i )

1,

1)

along the sections of the normal sheaf of) the proper transform of the axis"

ST (pos i ) ) ) l i n g on ( Ä (T))

The triangulated junior terahedron of the simplest example 1/5 (1,11,2) is drawn in fig. 10. The "low" tetrahedron induces the classical involutional terminal sin­gularit of t p e 1 / 2 1 )

basic simplex

non-basic, elementary

simplex

ig

Theor 4 ( E x e p t i o n ime divisors and intersection numbers).

Let ( C / G , [0]) = X (NG,AG) ,orb(ao)) denote the Gorenstein cyclic quotient singularity of type 8 . ) . If I satisfies condition 8. then :

(i) The exceptional locus of : X , Ä T) ) consists of ^

prime divisors

U) = V ( r » ( T « ; Star ( ( r » T ) ) ) 1 < ^

on T ) , with ffi>o ^ having the following structure :

ö O I - r - 1) as undles over F

for all T^T - l

LA if / 1 mod — 1)

if Z mod r - 1)

(ii) For the (higest intersection umers of tw consecutive exceptional prim divisors we get :

- D i + 1 l - r - l ) l))

Dj-D£1((r-l)-i 8.12)

fo th selfintersetio ers

8.1

for all T^T - l

r) if 1 mod r -

i—i

K J K J r - l)) if mod r -

i—

8.14)

Proof, (i) We distinguish three cases depending on the range of and the divisi bility condition on /

First ase. Let JZJ — 1 • Oviousl there are exactl two primi

tive collections in en Star ^ T) namel

)w i ) U ) and « 1 ) ) W ) «+1> ) 0 )

having no common elements. Furthermore

i i ) + n i + i ) 2 n « « 1 ) ) ü , ü+ 1) ) U ) U)

Hence each Dj has to e the total space of a P ^ u n d l e over an (r — 2)dimensional smooth compact toric variety, and since Star(r«;ÄG(T)) is a splitting fan it will b in addition, the total space of the projectivization of a decomposale

undle prop. 4.2 and thm. 4.4). On the other hand

en Star W r )

which means that Dj has Picard number 2, and has therefore to be isomorphic to the total space of the projectivization of a decomposable bundle of the form öp^2 (B O 2 ( ) over P^ (by Kleinschmidt's classification theorem 4.6). For this reason, it suffices to determine this single twisting number A by using lemma 4 . . It is eas to verif that we have the Z-linear dependence-relations

• • • i ((r - 1) + l)-l « / - r - 1) ) n « + 1 )

e.

£ ) ) - / - r - 1) ) n « ) )

onsequentl l ; l - l )

Second se. et / = 1 m o d r — 1) and JZJ ~~ Since

) U ) ) + ••• i ) W ) ) + ) ) „ , ^ , ,

we have oviousl ^

Third case. et I mod(r — 1) and j JZJ\ JZJ • Since rir equals lies on the facet conv({ei, . . . , e r_i}) of sG This means that the star of ^ in

T) can written as a direct product of the form

Stax ^ T ) Star ^ T) pos i)) a halfline in

and D splits into

p o s i ) ) ; Star j ) T) p o s i ) )

where this first factor is isomorphic to F because

ex-\ i r - 1) • n ( i )

ii) Since yi • Dj+1) yi Dj+

equals

J + ; S ( ( J + ;A ) U +

we o t a i n ( D 1 - D j + 1 ) l - r - l ) + l))

y (4.5). Similarly for (j • Dr^\); regarding Dj as the total space of a j bundle over P ^ , and making this time use of the "opposite" piece of the affine covering of its tp ica l f e r we get the second formula of 8.12) restricting

£l on Dj V G « ; Star ( ( « T ) ) )

and by (4.5) after sign-change (i.e., after having identified V ( T ^ + 1 ) )

with the divisor on D whose associated line undle is Ojjj (—1)).

On the other hand, O anc^ ^ n e adjunction formula gives

) ® °D

where DJ/X(NG,ÄG(T)) denotes the normal sheaf of Dj in ( jVG ,AG (T)) Hence, evaluating the highest power of the first Chern class of this sheaf at the fundamental c c l e Dj of Dj we tain for the selfintersection number :

= cr1 D = c 1 ) {[ = K^1 8.1

Formula 8.13) for D follows from 8.1) and 4 . ) ; formula 8.14) is ovious.

75

9. B a k i n g down he d e i n g u l a i z i n g m o h i s

The unique crepant resolution-morphism of the singularities discussed in the pr vious section can be factorized into (normalized) blow-ups in several w a s . give here two canonical decompositions of = fj of this kind

(a) Maintaining the notation introduced in the proof of thm. 8.2 for the construc tion of the unique, basic, coherent triangulation T esides s ( ; £ ) ' and c o n v a s ) for I 1 m o d r — 1)) we define

;{ j ) e 4 e ? e 4

as well as

;£ C O n v ; ^ ) )

and

;C j ' , . . . P ° S ; £ ) )

for all indices with

< -!—

v — 1

and for all r —1 possile r — 2)tuples Obviousl

s (j, j ; ^ s ; £

whenever + 1. Let now denote the positive integer

^^ + 1)J if modr~1) 5 ^ ) J + 1 if m o d r - l )

• For I = 1 mod ) we intoduce the following simplicial sd iv i s i ons of the junior simplex &

s ( 1 ) J s 1) ^ [ i S 2

conv ^ *=£ JET; & )

for all together with all their faces

and

1 "^ ' S1

for all ( ^ , ^ 2 , . . . ,£ r_ 2 ) G together with its faces

J^\ 731 lj?

for all together with all their faces

2 and alH, — 1.

nalogousl for I m o d r — 1) we define

1) Sr- ( 1 731 — 1)

7^1 ~ 1 731 ~~ 1) i;

r all together with all their faces

and

i l

for all (£i,?2>--- ;s>2) G together with its faces

+ l ; £ ) , ^ + l ^ j - - l ; £

^31 — 1 731 j J

for all together with all their faces

3 and ali — 2 and finall

I

l 2)

for all {^^•••^r2) G together with its faces

xt lemma is obvious constrction.

Lemma 9.1. All the above defined simplicial subdivisions Ti are triangu-lations and cover the entire G

P r o p o i o n 2 (First speedy factorization).

Let C/G (N, A o r b ) ) denote the Gorenstein cyclic quotient

singularity of type ) with I satisfying conditio 8.4). Then the iratio

resolution-morphism is the composite of the toric morphisms

(N i T)

=x(,A%)) , 1 < % < - 1 and T

In particular, in algebraic-geometric terms, one has

Xi+t = Norm [ rz

ef X;)] - 1

with centers

or <7o

P » * ' ) ) ^ 1 « < K - 2 ,

mod r — 1)

(pos ei _i})) mod 1)

Sketch of proof. Since NG is a "skew" lattice it is not so convenient to work directl with it For this reason we consider the linear transformation

y 1 y ) =M

wi • •

- 1

1)) • •

The image o the lattice via $ is the standard lattice Z l=i e - I* particular

+Zj ! - r - l ) ) | / - r - l ) ) Z e i

and $ maps this Z-basis o G onto

Q / - r - l ) ) T )

he p o s v e o h a n t onto the cone

ö ^ o - 0 pos

with

(ei) - l - l , - l I - r - 1)

\

and the Hubert basis (<70) o r t (6 )) onto

cr0)) ä^) c o n v e

} u «

where JZJ a n c

Ü) ^

For W The dual cone o Wo equals

ö^ ^ +l% +le?1l-r-l)) + le

(with { e ^ , . . , e } denoting the dual of {e i , . . . , e } ) . For every m belonging to the Hilbert basis )v CTÖ r t . the dual lattice ) define

^ I ( y ) < Mm ö ^ )

Suppose / 1 m o d r — 1) At first we shall show that for any fixed r — 2)-tuple with {q} = r - l we have

^ ( l ^ ) ) (91) r[+leV] r - 1

Moreover for I

te;f (92)

and finall pos a s ) ) I - r - 1)) + I e (9

Pof of (91) Suppose rst 1 bviousl ) a n d for all m Ö we get

and for ever

onversel let y denote an ab i t r a element o Write

as an M-linea combination w.rt . the basis {ei,. Since Ze 2 < < r — 1, and belong to <TÖ we obtain

y ) , < + i y ) y ) + erv,y) y)

e., ß M>o and therefore y € 1; 2 r — 1) pos({ )

Suppose now 2,. r — 1 Since

+ /e + i e e > f o r p - 2 , - l + l + 2

and m, (e^ +le^) , for all m ( (CTÖ)

the inclusion " c " is obvious Conversely, let y denote an bitrar element o +1 e\ and write it as Mlinea combination

5Z , ... l + l , ...

Since

+le + le y) <

and + / e y ) < + / < y ) ^

for p — 2 — l, + l, + 2 } we have

l ; 2 - l + l + 2r)) pos ({ U + u )

• Proof of (92) : Suppose first 1 The inclusion "C can be easily checked as before. Let y be an element o ^ and write it as linear combination

^ e i

have

y) < i=±

and

) < r - 1) y) 2 < < r - 1

as well as ( y ) < a - r - - l ) ) + / e y )

which is equivalent to

I - r) l - 1) / - r) ßr l ~ 1) ! - r

e . Hence,y £ a ( l = i ; 2 , r - l ) ) p o s ( { e i e r _ i ) .

• For # G { . . . , /• — 1} the proof of the inclusion " is again easy. To prove "D" it is enough to consider an element y r — 1) e and write it as linear combination

, . . . l + l , . . . l

definition satisfies the three inequalities

r - l K y ) < , y )

r _ 1 } ; _ r _ 1 ) } + j e

and r - 1) < , y > < r - 1) y)

for all — 1 + 1 + 2 irect evaluation combined with

/ - r - 1) + I e ) / - r - 1)) + Ze M ) I - r - 1)

gives K>o for alH 1 < < r e belongs to

( ± ± ; 2 - l + l pos (j eu U + u i

Prof of ( 9 ) : Since

I - r - 1)) +le I - (r - 1)) ^ +levr,e 2 < < r - 1

we have 0 ,. i 1 — r — 1)) + l e . On the other hand the definition of <TÖ

J - r - l ) + / e l < Vm ö^ ) .

onsquent

pos us) ) p o s ( { i l - r - l ) ) + l e

To show the converse inclusion take again a / — r — 1) +1 e] rite it as linear combination

l

i

and use the inequalities

l - r - l ) + l e y )

l - r - l ) ) e + l e y ) y ) r-l)

and for all j 2 < j < r — 1 the inequalities

J - r - 1)) e + Ze r - 1) r - 1)

(91) (92) (9 and lemma 1 we obtain

_, $ 1 (r [m]) together 1 ^ with all their faces

e * H l b i

This means that A Q ( T I ) = (Ag) b l [orb (To)] by proposition 3.8, and therefore gi is indeed the proper birational morphism corresponding to the normalized usual blow-up o (NQ, Q) at the closed point orb(ao).

• If K > 2, in the second step we blow up (simultaneousl) the r — 2)-dimensional common sinular locus o all ne charts U i- \. ote that locall

a neighborhood o uch a singular point within T I can be viewed

like a 2-dimensional Ai- sinularit j

To prove that the above defined triangulation 12 induces the normalization o this blow-up one applies theorem 3.6 and techniques similar to those used for Ti. The details are left as an exercise to the reader. Repeating the described procedre altogether — 1 times we rive at the entire basic triangulation T = X«

• The proof in the case in which / = 0 mod(r — 1) can be done analogously and will be omitted. The only difference is that in the last step one blows up (once) the remaining 1-dimensional singular locus pos ({ei i ) ) inherited from the single non-basi facet o

Figures 11 a) and (b) show this speedy factorization of / for the singularities o pe 1/1 18 ) and 1/11 respectivel

82

g

(a) (b)

Fie

(b) A second canonical factorization of / = fj is c o n s t c t e d means o the followin ^~f riangulations o the junior simplex:

(l;£ c o n v ( j W e 4 e ?

for all ( together with all their faces

and

+l

c o n v ( { £ l £

for all (£i,£2>--->£r2) e together with its faces

+ l;^ c o n v + 1 )

r all together with all their aces

for alH 1 < — 1

P r o p o i o n 3 ( o n factorization).

Let (V /G, [ (X (NQ, A ß ) , orb (00)) be the Gorenstein cyclic quotient singu­larity of type with I satisfying condition (8.4). Tien the birational resolution-

morphism can be expressed also as the composite of F~T toric morphisms

( N ) ^ X l ^ ^ ! ^ T)

Xi : - 1 and trival triangulation T.

In particular,

Xi+1 ^ orm g pQ)l , - 1

for appropriate Öideal sheaves T such that upp /%), where

or i =

p o s ) )

either ,1 <i <

and / 1 mod r — 1)

v < v —

and Z mod r — 1)

pos ({ )) I E mod r - 1) — \

84

(Up to the above last case and the case in which all Z ' s are endowed with a non-reduced scheme s t c t u r e )

Moreover, one has

+1 orm [ 9 | ^ S t a (f ) )

with respect to the different lattice

H Zn Ze - - - Z e i

and the star of the cone

either — 1,

and I 1 mod r — 1)

({ r al wi i/ < v

and / mod r — 1)

were an (g entes th cyclic group of analtic automorpism of

% ; Star ( f ) )

generated by

i'- ) t i l ) i l ) l i i )

The roof o 9.3 is an immediate generalization o that of the case in which r = (see 7.2 (ii)) relies on a successive application of and is left as an exercise to the reader The only difference is that whenever r > 3 and I = 0 m o d r — 1) we also blow p the remaining 1-dimensional singular locus in the last step). Figures 12 (a) and (b) illustrate the triangulations inducing the factorization of / for the singularities o e 1/6 ( 4 ) and 1/ r e s c t i v e l

jh1 lh1

(a)

Fi

Remark 9.4. (i) Combining propositions 2 and 9.3 with corolla 4 one ma obtain alternative roofs o the projectivity of / fj

(ii) The proofs o 9. and 9.3 work after minor modifcations even i one omits the assumtion for T to be basi

Exercise 9.5. Under the assumtion o ro determine a single O x G y ideal sheaf X uch that Sing(AT (NG, A^)) p p ( ö / ^ ) and / r itsel is nothin ut the normalized blow-

orm , A ( N / G

(N f. thm 3 1 and rem iii))

10. Further remar and a conjecture

As we already saw in 7.8 (ii), in dimension three, besides j (1,1,1 — 2)'s there is also another "new" Gorenstein, cli quotient singularity having a unique, pro­jective, crepant resolution namel j 4) This can be generalized in arbitrar dimensions too

Theorem 10.1. The cyclic Gorenstein quotient singularity of type

^—^ (l

can he fully solved by a torus-equivariant projective crepant mhism in all dimensions r 2 Moreover, up to isomorphism, this resolution is unique

The proof of theorem 10.1 will be given in [13] As you guess, the required trian-ulation T will be the high-dimensional analogue of that of figure 9. The details

of the proof of the uniqueness of T and of the fact that it is indeed basic are somewhat lengthy, and involve binary representations, explicit Hilbert-basis de termination and some tricks with determinants. The coherence of T, on the other hand, can be shown directly by using tools from the theory o l t o e s e avoiding both atching lemma and factorization arguments

Forgetting completel the uniqueness-property, we believe that this single singu­larit is again nothing but "the first member" o an infinite family of Gorenstein cyclic quotient singularities (10.1) called for simlicit r-dimensional geometric progress singularity-series of ratio k in notation: GPSSr; k) all o whose mem­bers admit the desired resolutions

Conjecture 1 0 2 ( r ; fc)one) ll cyclic orenstein tien sin gularities of type

1 -

F (i "

U 1)

mit toseivaria projetiv crept full resolutio for all d all 2

Exercise 10. As a first pproach to 1 eg . to PSS(4;fc)-conjecture) con­sider the example ^ 9 , ) with k = 3) and normalize the blow-up X (NG,AG) at orb((7o) (equipped with the reduced s t ructre) . What kind o triangulation of the junior tetrahedron will be induced this procedure ? What would you expect as "next s t e " ? int Relate what "see" with the singularities being studied in 12

cknowledgements. The first author would like to express his thanks to I Naka-mura and M Reid for having pointed out that the existence o rojective crepant resolutions o C jG in dimensions r > 4 does not necessarily imp the smoothness f the corresponding Hilbert scheme of G-orbits to . Batyrev for a discussion

about the structure o the how ring A (P(£)), to G. M. Ziegler for an e-mail about the counterexample 6.7 (ii), to DFG for the support by an one-yearresearch-fellowshi and to Mathematics Institute o Bonn Universit for hositali t until the end o March, 1

The second author thanks R Firla for explaining to him his counterexample 16 (i), DFG for the support the Leibniz-Preis awarded to M.Grötschel, and

Konrad-ZuseZentrum in Berlin for ideal workin conditions d r i n g the writing o this er

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