论文标题
在Behrend功能和某些脂肪点的爆炸
On the Behrend function and the blowup of some fat points
论文作者
论文摘要
$ \ Mathbb c $ -scheme $ x $的behrend函数是一个构造函数$ν_x\ colon x（\ mathbb c）\ to \ mathbb z $由behrend引入的\ mathbb z $，对$ x $的方案结构的内在性。它是$ x $的奇异性（微妙的）不变，在枚举几何学中扮演着重要角色。迄今为止，仅知道behrend函数的少数一般属性。在本文中，我们将其计算为大量脂肪点（单点支持的方案）。我们首先要观察到，如果$ x \ hookrightArrow \ mathbb a^n $是一个胖点，则$ν_x$是异常的divisor $ e_ {x} \ mathbb a^n $在blowup $ \ textrm {bl} bl} _ {bl} _ {x} x} x}^b afordifiate divisor $ e_ {x} \ mathbb a^n $的多样性的总和。此外，我们证明$ν_x$可以通过$ \ textrm {bl} _ {x} \ mathbb a^n $的标准化来明确计算。 我们明确的公式证明了$ \ mathbb a^2 $中脂肪点的behrend功能的证明很大程度上取决于曲折的几何技术。一路上，我们找到了$ e_ {x} \ mathbb a^2 $的不可约组件数量的公式，其中$ x \ jookrightArrow \ mathbb a^2 $是一个胖点，使得$ \ textrm {bl} _ {x} _ {x}} \ mathbb a^2 $正常。
The Behrend function of a $\mathbb C$-scheme $X$ is a constructible function $ν_X\colon X(\mathbb C) \to \mathbb Z$ introduced by Behrend, intrinsic to the scheme structure of $X$. It is a (subtle) invariant of singularities of $X$, playing a prominent role in enumerative geometry. To date, only a handful of general properties of the Behrend function are known. In this paper, we compute it for a large class of fat points (schemes supported at a single point). We first observe that, if $X \hookrightarrow \mathbb A^N$ is a fat point, $ν_X$ is the sum of the multiplicities of the irreducible components of the exceptional divisor $E_{X}\mathbb A^N$ in the blowup $\textrm{Bl}_{X}\mathbb A^N$. Moreover, we prove that $ν_X$ can be computed explicitly through the normalisation of $\textrm{Bl}_{X}\mathbb A^N$. The proofs of our explicit formulas for the Behrend function of a fat point in $\mathbb A^2$ rely heavily on toric geometry techniques. Along the way, we find a formula for the number of irreducible components of $E_{X}\mathbb A^2$, where $X \hookrightarrow \mathbb A^2$ is a fat point such that $\textrm{Bl}_{X}\mathbb A^2$ is normal.
