AI 中文总结
本文研究零维概型的Behrend函数计算问题，以爆破代数为主要工具，得到零维单项式概型的显式公式，推广了Graffeo--Ricolfi的工作，同时建立了相关独立结果。
AI 中文摘要
给定复数域上有限型概型$X$，Behrend函数是Behrend引入的可构造函数$\nu_X: X(\boldsymbol{C}) \rightarrow \boldsymbol{Z}$，用于定义Donaldson--Thomas理论中的计数不变量。即使在简单情形下，Behrend函数也极难计算。本文研究零维概型的Behrend函数计算问题，得到多个显式公式，尤其针对任意零维单项式概型，大幅推广了Graffeo--Ricolfi的前期工作。本文的主要工具来自爆破代数理论，同时建立了与整数分解性质、加权Veronese子环及约化纤维环相关的独立有意义结果。
英文摘要
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.
