Polynomiality of the double ramification cycle

TL;DR

This paper provides an alternative proof that the double ramification cycle depends polynomially on the input data, confirming a long-standing conjecture in algebraic geometry.

Contribution

It offers a new proof demonstrating the polynomial dependence of the double ramification cycle on the data A, advancing understanding of its structure.

Findings

Confirmed polynomial dependence of the double ramification cycle on A

Provided an alternative proof to existing results

Strengthened the theoretical foundation of the cycle's properties

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 $(\omega_C^{\log})^{-k}\left(\sum a_i p_i\right)$ 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.