Characteristic numbers of algebras

TL;DR

This paper introduces characteristic numbers as numerical invariants of finite commutative unital complex algebras, characterizing special classes and analyzing their behavior on Hilbert schemes.

Contribution

It defines characteristic numbers for algebras, characterizes Gorenstein and complete intersection algebras via these numbers, and studies their bounds on Hilbert schemes.

Findings

Characteristic numbers are constant on                 

Bounds for characteristic numbers on Hilbert schemes are explicitly provided for smoothable and Gorenstein loci.

The paper characterizes algebra classes using characteristic numbers and computes these for specific algebra families.

Abstract

We introduce characteristic numbers of a finite commutative unital $\mathbb{C}$-algebra, which are numerical invariants arising from algebraic intersection theory. We characterize Gorenstein and local complete intersection algebras in terms of their characteristic numbers. We compute characteristic numbers for certain families of algebras. We show that characteristic numbers are constant on $\mathrm{Hilb}_d(\mathbb{A}^1)$, provide an explicit upper bound for characteristic numbers on the smoothable component of $\mathrm{Hilb}_d(\mathbb{A}^n)$ and an explicit lower bound for characteristic numbers on the Gorenstein locus of $\mathrm{Hilb}_d(\mathbb{A}^n)$ for $n \geq d-2$.