Quotients of Jacobians

TL;DR

This paper studies the geometric properties of quotients of Jacobian varieties of algebraic curves under automorphism groups, establishing conditions for canonical singularities and uniruledness based on genus.

Contribution

It proves that for genus greater than 20, the quotient JC/G has canonical singularities and Kodaira dimension zero, and provides examples for lower genus where JC/G is uniruled.

Findings

For g > 20, JC/G has canonical singularities.

For g < 5, JC/G can be uniruled.

The paper links genus to geometric properties of Jacobian quotients.

Abstract

Let C be a curve of genus g, and G a finite group of automorphisms of C . We prove that for g > 20 the quotient JC/G has canonical singularities, hence Kodaira dimension 0. On the other hand we give examples of curves C with g < 5 for which JC/G is uniruled.