Konrad-zuse-zentrum F ¨ Ur Informationstechnik Berlin on a Series of Gorenstein Cyclic Quotient Singularities Admitting a Unique Projective Crepant Resolution on a Series of Gorenstein Cyclic Quotient Singularities Admitting a Unique Projective Crepant Resolution
Konrad-zuse-zentrum F ¨ Ur Informationstechnik Berlin on a Series of Gorenstein Cyclic Quotient Singularities Admitting a Unique Projective Crepant Resolution on a Series of Gorenstein Cyclic Quotient Singularities Admitting a Unique Projective Crepant Resolution
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Konrad-zuse-zentrum F ¡ Ur Informationstechnik Berlin 论一系列 Gorenstein 环商奇点 承认唯一的射影 Crepant 分辨率 关于一系列 Gorenstein 循环商奇点 承认唯一的射影 Crepant 分辨率
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通讯作者：
M. Henk
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作者：
D. Dais;M. Henk
设G是SL(r；C)的nite子群。在维度r=2和r=3中，McKay对应提供了G的IR-可约表示集与C_r=G的射影、Crepant去奇化的覆盖空间的上同调-环基之间的自然双射。对于r=2，这种去奇化是唯一的，并且已知由G-轨道的Hilbert格式决定。类似的陈述(包括在C3=G的所有可能的光滑极小模型中只区分一个的方法)，很可能对所有G的SL(3；C)也是正确的，最近由于Ito，Nakamura和Reid而产生的Hilbert格式技巧，有望导致一个新的迷人的一致理论。然而，对于尺寸r 4，要应用类似的技术，需要额外的修改。此外，C r=G的极小模型只有在特殊情况下才是光滑的。例如，C4=(对合)不可能有任何光滑的极小模型。另一方面，所有是C.I.S的交换商空间总是可以用环面等变的、可折的、射影态射完全分解的。因此，从一开始，一个给定的Gorenstein商空间Cr=G，r4，是否允许这种特殊的奇异表示的问题似乎是绝对关键的。在本文中，我们从环面几何的角度简要地介绍了这种去奇异的存在性问题，证明了第一类Gorenstein循环商奇性是L(1；：：；1；L？(r？1))对于L r2，有一个唯一的环面等变射影的正解，它是完全的“II”，或者L 0 mod(r？1)或L 1 mod(r？1).结果表明，如果这两个条件中的一个是满的，则完全去奇化的例外轨迹由L r？1素因子组成，L r？1？1与Pr？2C上的P1C-丛的全空间同构.此外，证明了交数是可显式计算的，并且分解态射可以看作是连续的(正规的)爆破的合成.显然，上述类型的单参化奇点级数包含(作为其第一个成员)著名的Gorenstein奇点，该奇点由位于Pr？1C的r-字节组Veronese嵌入上的Aane锥的原点来表示。
Let G be a nite 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 r =G. 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 C 3 =G), are very probably true for all G's 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 modiications. In addition, minimal models of C r =G are smooth only under special circumstances. C 4 = (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 r =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 1 l (1; : : : ; 1; l ? (r ? 1)) with l r 2, have a unique torus-equivariant projective, crepant, partial resolution , which is \full" ii either l 0 mod (r ? 1) or l 1 mod (r ? 1). As it turns out, if one of these two conditions is fulllled, then the exceptional locus of the full desingularization consists of l r?1 prime divisors, l r?1 ? 1 of which are isomorphic to the total spaces of P 1 C-bundles over P r?2 C. 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 \\rst member") the well-known Gorenstein singularity deened by the origin of the aane cone which lies over the r-tuple Veronese embedding of P r?1 C .