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 $\alpha_{ \mathcal A,\mu}$.
Categories: math.AG