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Euler characteristics of higher rank double ramification loci in genus one

Luca Battistella, Navid Nabijou

TL;DR

This work computes the orbifold Euler characteristics of genus-one double ramification loci and their higher-rank generalisations, yielding explicit closed formulas. The rank-one case produces a polynomial in the sum of squared ramification data, while the higher-rank case expresses χ_{ ext{orb}} as a gcd-of-minors sum over contractions of the DR matrix, with a guiding recurrence. The proofs hinge on a forgetful recursion in genus one, stratified étale analysis, and GL_r(Z) invariance to reduce to special forms, enabling induction on the rank and number of markings. The results are rooted in intersection-theoretic interpretations and connected to a broader all-genus Hurwitz-stratification program, with computational verification and potential normal-crossings compactifications discussed as future directions.

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.

Euler characteristics of higher rank double ramification loci in genus one

TL;DR

This work computes the orbifold Euler characteristics of genus-one double ramification loci and their higher-rank generalisations, yielding explicit closed formulas. The rank-one case produces a polynomial in the sum of squared ramification data, while the higher-rank case expresses χ_{ ext{orb}} as a gcd-of-minors sum over contractions of the DR matrix, with a guiding recurrence. The proofs hinge on a forgetful recursion in genus one, stratified étale analysis, and GL_r(Z) invariance to reduce to special forms, enabling induction on the rank and number of markings. The results are rooted in intersection-theoretic interpretations and connected to a broader all-genus Hurwitz-stratification program, with computational verification and potential normal-crossings compactifications discussed as future directions.

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.

Paper Structure

This paper contains 13 sections, 15 theorems, 96 equations.

Table of Contents

  1. Results
  2. Proof strategy
  3. Intersection theory
  4. Higher genus: the Hurwitz stratification
  5. Rank one
  6. Higher rank
  7. Formula
  8. Reduction via $\operatorname{GL}_r(\mathbb{Z})$ invariance
  9. Geometric recursion
  10. Matrix lemmas
  11. Proof
  12. Simplifying the leading term
  13. Comparing Theorems \ref{['thm: Euler char rank one']} and \ref{['thm: higher rank DR']}

Key Result

Theorem X

Given $a=(a_1,\ldots,a_n)$ the orbifold Euler characteristic of $\operatorname{DR}_{1,n}(a)$ is given by:

Theorems & Definitions (29)

  • Theorem X: \ref{['thm: Euler char rank one']}
  • Theorem Y: \ref{['thm: higher rank DR']}
  • Theorem Z: \ref{['thm: recursion in Grothendieck ring']}
  • Theorem 1.1: \ref{['thm: rank one introduction']}
  • Theorem 1.2: \ref{['thm: recursion introduction']}
  • proof
  • proof : Proof of \ref{['thm: Euler char rank one']}
  • Lemma 1.3: Harer--Zagier in genus one
  • proof
  • Remark 2.1
  • ...and 19 more