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 ${\nu }_X:X(\mathbb{C})\to \mathbb{Z}$ 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 ${\text{Bl}}_X{\mathbb{A}}^N$. Moreover, we prove that ${\nu }_X$ can be computed explicitly through the normalisation of ${\text{Bl}}_X{\mathbb{A}}^N$. 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 ${\text{Bl}}_X{\mathbb{A}}^2$ is normal.

53 pages, comments welcome!