A log resolution for the theta divisor of a hyperelliptic curve

TL;DR

This paper constructs an explicit resolution of singularities for the theta divisor of a hyperelliptic curve using iterated blowups, and shows the Brill-Noether stratification forms a Whitney stratification.

Contribution

It provides a natural, explicit embedded resolution of the theta divisor's singularities for hyperelliptic curves, advancing understanding of their geometric structure.

Findings

Explicit embedded resolution via iterated blowups

Brill-Noether stratification is Whitney stratification

Enhanced understanding of hyperelliptic Jacobian geometry

Abstract

In this paper, we prove that the theta divisor of a smooth hyperelliptic curve has a natural and explicit embedded resolution of singularities using iterated blowups of Brill-Noether subvarieties. We also show that the Brill-Noether stratification of the hyperelliptic Jacobian is a Whitney stratification.