On the Behrend function and the blowup of some fat points

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.