Maximal ideal cycles over normal surface singularities of Brieskorn type
Maximal ideal cycles over normal surface singularities of Brieskorn type
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DOI：
10.18910/4341
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发表时间：
2012-03
影响因子：
0.4
通讯作者：
K. Konno;D. Nagashima
中科院分区：
文献类型：
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作者：
K. Konno;D. Nagashima
对于正规的二维Brieskorn型超曲面奇点，给出了星形好分辨空间上的基本圈和最大理想圈的具体刻画。确定这两个周期何时重合。引言设(V，o)是正规曲面奇点的芽，W(X，E)！(v，o)一种归结，其中E D 1(O)表示例外集.设E D SriD1Ei是E的不可约分解，形式和YDPriD1IEi(I2Z)称为E上的一个圈，对于一个循环Y，如果YEi对所有i都是0，则称Y在E上是NEF的.由于交形式在E上是负定的，集合{Y0 jY在E上是nef}是非空的，且有最小的元素ze，E上的基本圈称为(V，o)的基本亏格，我们记为p f(V，o).设m是OV的极大理想，o.对于任意非零f2m，f AE的零因子可记为(FAE)D(FAE)XCD，其中(FAE)X是E上的一个圈，D是一个不涉及任何Ei‘S的有效因子.我们称(FAE)X为E上由f2m领导的圈.理论纤维o的除数部分ME称为E上的最大理想圈.如果f1，：：，f2m生成m，然后ME D inf1 I(Fi Ae)X by[14，命题2.12]。因为我不是，所以我们总是没有我。对于有理奇点、Kodaira奇点和{zn Df(x，y)}(n2，f2 C{x，y})型奇点，当n为0时，有时会出现ME D ze。对于最后一种类型，Tomaru在[10，定理4.1]中证明了当n除以阶数(F)时，两个圈在任意分解上重合，推广了Dixon[2，定理1]中关于n-D2的著名结果。然而，即使对于一类特殊的奇点，也需要更系统的研究，以澄清这种重要周期的重合发生在什么时候。2010年《数学学科分类》。小学14J17；中学32S25。226K.Konno和D.Nagashima
For normal two dimensional hypersurface singularities of Brieskorn type, concrete descriptions are given to both the fundamental cycle and the maximal ideal cycle on a star-shaped good resolution space. It is determined when these two cycles coincide. Introduction Let (V , o) be a germ of a normal surface singularity and W (X, E) ! (V , o) a resolution, where E D 1(o) denotes the exceptional set. Let E D SriD1 Ei be the irreducible decomposition of E. A formal sum Y D PriD1 i Ei ( i 2 Z) is called a cycle on E. For a cycle Y , Y is said to be nef on E if Y Ei 0 for all i . Since the intersection form is negative definite on E, the set {Y 0 j Y is nef on E} is nonempty and has the smallest element ZE, the fundamental cycle on E. The arithmetic genus of ZE is called the fundamental genus of (V , o) and we denote it by p f (V , o). Let m be the maximal ideal of OV ,o. For any non-zero f 2 m, the zero divisor of f AE can be written as ( f AE ) D ( f AE )X C D, where ( f AE )X is a cycle on E and D is an effective divisor which does not involve any of Ei ’s. We call ( f AE )X the cycle on E led by f 2 m. The divisorial part ME of the scheme theoretic fiber o is said to be the maximal ideal cycle on E. If f1, ::: , f 2 m generate m, then ME D inf1 i ( fi AE )X by [14, Proposition 2.12]. Since ME is nef, we always have 0 ZE ME. It sometimes happens that ME D ZE, as one can observe for rational singular points, Kodaira singular points and singularities of type {zn D f (x , y)} (n 2, f 2 C{x , y}) when n 0. As for the last type, Tomaru proved in [10, Theorem 4.1] that two cycles coincide on any resolution when n divides ord( f ), extending the well-known result for n D 2 due to Dixon [2, Theorem 1]. However, even for a particular class of singularities, a more systematic study will be required in order to clarify when such a coincidence of important cycles occurs. 2010 Mathematics Subject Classification. Primary 14J17; Secondary 32S25. 226 K. KONNO AND D. NAGASHIMA