Polynomiality of the double ramification cycle

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.

12 pages. Comments very welcome!