# Divisor varieties of symmetric products

## TL;DR

This paper investigates the geometry of divisor varieties on symmetric products of algebraic curves, extending classical Brill-Noether theory to higher dimensions and providing detailed insights into their structure.

## Contribution

It offers a detailed study of divisor varieties on symmetric products of curves, advancing understanding beyond classical curve theory into higher-dimensional cases.

## Key findings

- Describes the structure of divisor varieties on symmetric products
- Provides conditions for smoothness and irreducibility
- Extends Brill-Noether theory to higher dimensions

## Abstract

The geometry of divisors on algebraic curves has been studied extensively over the years. The foundational results of this Brill-Noether theory imply that on a general curve, the spaces parametrizing linear series (of fixed degree and dimension) are smooth, irreducible projective varieties of known dimension. For higher dimensional varieties, the story is less well understood. Our purpose in this paper is to study in detail one class of higher dimensional examples where one can hope for a quite detailed picture, namely (the spaces parametrizing) divisors on the symmetric product of a curve.