Divergent Series and Serre's intersection formula for graded rings

TL;DR

This paper explores how to assign meaningful intersection multiplicities in singular algebraic varieties by relating divergent series from Serre's intersection formula to fractional multiplicities through analytic continuation, specifically in graded rings.

Contribution

It demonstrates a positive connection between divergent series in Serre's formula and fractional intersection multiplicities using analytic continuation in the context of graded rings.

Findings

Established a link between divergent series and fractional multiplicities.

Applied Avramov and Buchweitz's work to graded rings.

Provided a positive answer to Fulton's question.

Abstract

On a smooth variety, Serre's intersection formula computes intersection multiplicities via an alternating sum of the lengths of Tor groups. When the variety is singular, the corresponding sum can be a divergent series. But there are alternate geometric approaches for assigning (often fractional) intersection multiplicities in some singular settings. Our motivating question comes from Fulton, who asks whether an analytic continuation of the divergent series from Serre's formula can be related to these fractional multiplicities. By applying work of Avramov and Buchweitz, we positively answer Fulton's question in the context of graded rings.