Behrend function and blowup algebras

Abstract

Given a scheme $X$ of finite type over the complex numbers, the Behrend function is a constructible function $ν_X: X(\mathbb C) \rightarrow \mathbb Z $ introduced by Behrend in order to define enumerative invariants in Donaldson--Thomas theory. Even in simple cases, the Behrend function is very difficult to compute. In this article, we tackle the problem of computing the Behrend function of zero-dimensional schemes. We obtain a number of explicit formulas, in particular, for arbitrary zero-dimensional monomial schemes, thus providing vast generalizations of previous work of Graffeo--Ricolfi. Our main tools come from the theory of blowup algebras. Along the way, we establish results of independent interest related to the integer decomposition property, weighted Veronese subrings, and reduced fiber rings.