There is no degree map for 0-cycles on Artin stacks

TL;DR

This paper proves that it is impossible to define a consistent degree map for 0-cycles on certain Artin stacks that aligns with natural properties like non-zero degrees for points and compatibility with closed immersions.

Contribution

It establishes a fundamental obstruction to defining degree maps for 0-cycles on Artin stacks with proper good moduli spaces.

Findings

No degree map can assign non-zero degrees to points while remaining compatible with closed immersions.

The result highlights limitations in extending classical cycle degree concepts to Artin stacks.

Provides insight into the structure of 0-cycles on algebraic stacks and their invariants.

Abstract

We show that there is no way to define degrees of 0-cycles on Artin stacks with proper good moduli spaces so that (i) the degree of an ordinary point is non-zero, and (ii) degrees are compatible with closed immersions.