# Moving Seshadri Constants, and Coverings of Varieties of Maximal Albanese Dimension

## TL;DR

The paper demonstrates that for varieties of maximal Albanese dimension, one can find finite abelian covers where the moving Seshadri constants of pulled-back line bundles become arbitrarily large, improving geometric properties like jet separation.

## Contribution

It establishes the growth of moving Seshadri constants on covers of maximal Albanese dimension varieties and applies this to enhance the separation of jets and control of the Albanese map's exceptional locus.

## Key findings

- Moving Seshadri constants can be made arbitrarily large on suitable covers.
- Existence of covers where the adjoint system separates any number of jets.
- Control over the exceptional locus of the Albanese map via covers.

## Abstract

Let $X$ be a smooth projective complex variety of maximal Albanese dimension, and let $L \to X$ be a big line bundle. We prove that the moving Seshadri constants of the pull-backs of $L$ to suitable finite abelian \'etale covers of $X$ are arbitrarily large. As an application, given any integer $k\geq 1$, there exists an abelian \'etale cover $p\colon X' \to X$ such that the adjoint system $\big|K_{X'} + p^*L \big|$ separates $k$-jets away from the augmented base locus of $p^*L$, and the exceptional locus of the pull-back of the Albanese map of $X$ under $p$.