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Polynomiality of the double ramification cycle

Pim Spelier

Abstract

Let $A = (a_1,\dots,a_n)\in \mathbb{Z}^n$ be a sequence with sum $k(2g-2+n)$. The double ramification cycle $\mathsf{DR}_g(A) \in \mathsf{CH}^g(\bar{\mathcal{M}}_{g,n})$ is the virtual class of the locus of curves $(C,p_1,\dots,p_n)$ where the line bundle $(ω_C^{\log})^{-k}\left(\sum a_i p_i\right)$ is trivial. Although there has long been a formula for $\mathsf{DR}_g(A)$ [JPPZ17], the exact dependence on $A$ was unknown for a long time, though it was conjectured to be polynomial in $A$. A proof was announced in [JPPZ17], and Pixton gave a proof incorporating ideas of Zagier in [Pix23]. Here we present an alternative proof of the polynomiality of the double ramification cycle.

Polynomiality of the double ramification cycle

Abstract

Let be a sequence with sum . The double ramification cycle is the virtual class of the locus of curves where the line bundle is trivial. Although there has long been a formula for [JPPZ17], the exact dependence on was unknown for a long time, though it was conjectured to be polynomial in . A proof was announced in [JPPZ17], and Pixton gave a proof incorporating ideas of Zagier in [Pix23]. Here we present an alternative proof of the polynomiality of the double ramification cycle.
Paper Structure (8 sections, 8 theorems, 42 equations)

This paper contains 8 sections, 8 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Definitions
  3. Pixton's formula for the double ramification cycle
  4. Sums over weightings
  5. Separating edge case
  6. Non-separating case
  7. Proof of Theorem 4.1
  8. Polynomiality of the double ramification cycle

Key Result

Theorem A

Fix $g,n$. The cycle $\operatorname{DR}_g(a_1,\dots,a_n) \in \mathop{\mathrm{CH}}\nolimits^g(M_{g,n})$ is a polynomial in $(a_1,\dots,a_n) \in \mathbb{Z}^n$, where we require that $(2g-2+n)\mid \sum a_i$.

Theorems & Definitions (20)

  • Theorem A: \ref{['thm:drpoly']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4: Janda2016Double-ramifica
  • Definition 2.5
  • Theorem 4.1
  • Remark 4.2
  • Lemma 4.3
  • proof
  • ...and 10 more