# The direct image of generalized divisors and the Norm map between compactified Jacobians

## TL;DR

This paper develops a framework for direct and inverse images of generalized divisors and line bundles over schemes, introduces Norm and inverse image maps between compactified Jacobians of curves, and studies the Prym stack as their kernel.

## Contribution

It defines new notions of direct and inverse images for generalized divisors and line bundles, and explores the properties of Norm maps and Prym stacks in the context of compactified Jacobians.

## Key findings

- Defined direct and inverse image for generalized divisors.
- Introduced and analyzed Norm and inverse image maps between compactified Jacobians.
- Studied the Prym stack as the kernel of the Norm map.

## Abstract

Given a finite, flat morphism between embeddable noetherian schemes of pure dimension 1, we define the notion of direct and inverse image for generalized divisors and generalized line bundles. In the case when we deal with (possibly reducible, non-reduced) projective curves over a field and the codomain curve is smooth, we introduce the compactified Jacobians parametrizing torsion-free rank-1 sheaves and we study the Norm and inverse image maps between compactified Jacobians. Finally, we introduce and study the Prym stack defined as the kernel of the Norm map.