On the Behrend function and the blowup of some fat points

TL;DR

This paper computes the Behrend function for a broad class of fat points in algebraic geometry, revealing its relation to blowups and normalization, and employs toric geometry techniques for explicit calculations.

Contribution

It provides explicit formulas for the Behrend function of fat points using blowup and normalization, advancing understanding of this invariant in singularity theory.

Findings

Behrend function equals sum of multiplicities of exceptional divisor components

Explicit computation of Behrend function via normalization of blowup

Formula for the number of irreducible components of the exceptional divisor

Abstract

The Behrend function of a $\mathbb C$-scheme $X$ is a constructible function $\nu_X\colon X(\mathbb C) \to \mathbb Z$ 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, $\nu_X$ 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 $\nu_X$ 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.