Euler characteristics of higher rank double ramification loci in genus one

TL;DR

This paper derives explicit formulas for the orbifold Euler characteristics of higher-rank double ramification loci in genus one, extending previous results and providing new computational tools for these moduli spaces.

Contribution

It introduces closed-form formulas for the Euler characteristics of higher-rank loci, generalizing known rank-one results and employing a recurrence relation for their computation.

Findings

Rank-one formula is a polynomial

Higher-rank formula involves gcd of matrix minors

Recurrence relation enables induction on rank and markings

Abstract

Double ramification loci parametrise marked curves where a weighted sum of the markings is linearly trivial; higher-rank loci are obtained by imposing several such conditions simultaneously. We obtain closed formulae for the orbifold Euler characteristics of double ramification loci, and their higher-rank generalisations, in genus one. The rank-one formula is a polynomial, while the higher-rank formula involves greatest common divisors of matrix minors. The proof is based on a recurrence relation, which allows for induction on the rank and number of markings.