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

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 $\overline{\mathcal M}<em>{g,n+1}$ be the moduli space of $(n+1)$-pointed stable curves of genus $g$ and $$\overline{\mathcal J}</em>{\mu,g}\rightarrow \overline{\mathcal M}<em>{g,1}$$ 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 $$\alpha</em>{\mathcal A,\mu}\colon \overline{\mathcal M}<em>{g,n+1}\dashrightarrow \overline{\mathcal J}</em>{\mu,g}$$, taking a pointed curve $(X,p_0,\dots,p_n)$ to $\mathcal{O}<em>X(\sum</em>{0\le i\le n} a_ip_i)$. The blowup of $\overline{\mathcal M}<em>{g,n+1}$ giving rise to the resolution is inspired by the resolution of the tropical analogue of the map $\alpha</em>{\mathcal A,\mu}$ (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 $\alpha_{\mathcal A,\mu}$.