The resolution of the universal Abel map via tropical geometry and applications

TL;DR

This paper provides an explicit resolution of the universal Abel map using tropical geometry techniques and applies this to describe the double ramification cycle in algebraic geometry.

Contribution

It introduces a new explicit resolution of the universal Abel map via tropical geometry and relates it to the double ramification cycle.

Findings

Explicit resolution of the universal Abel map constructed.

Resolution inspired by tropical analogue in generalized cone complexes.

Application to describing the double ramification cycle.

Abstract

Let $g$ and $n$ be nonnegative integers and $\mathcal A=(a_0,\dots,a_n)$ a sequence of $n+1$ integers summing up to $d$. Let $\overline{\mathcal M}_{g,n+1}$ be the moduli space of $(n+1)$-pointed stable curves of genus $g$ and $\overline{\mathcal J}_{\mu,g}\rightarrow \overline{\mathcal M}_{g,1}$ be the Esteves' universal Jacobian, where $\mu$ is a universal genus-$g$ polarization of degree $d$. We give an explicit resolution of the universal Abel map $\alpha_{\mathcal A,\mu}\colon \overline{\mathcal M}_{g,n+1}\dashrightarrow \overline{\mathcal J}_{\mu,g}$, taking a pointed curve $(X,p_0,\dots,p_n)$ to $\mathcal{O}_X(\sum_{0\le i\le n} a_ip_i)$. The blowup of $\overline{\mathcal M}_{g,n+1}$ giving rise to the resolution is inspired by the resolution of the tropical analogue of the map $\alpha_{\mathcal A,\mu}$ (in the category of generalized cone complexes). As an application, we describe the double ramification cycle in terms of the universal sheaf inducing the resolution of the map $\alpha_{\mathcal A,\mu}$.

Figures