# A Density Result for Real Hyperelliptic Curves

## TL;DR

This paper proves that for a dense set of parameters, a specific divisor on real hyperelliptic curves is torsion in the Jacobian, using degeneration and period map analysis.

## Contribution

It establishes the density of parameters for which a divisor is torsion in the Jacobian of real hyperelliptic curves, via degeneration to nodal curves and period map analysis.

## Key findings

- Divisor $([
abla^+] - [
abla^-])$ is torsion for dense parameters
- Period map derivative is generically full rank
- Degeneration approach confirms torsion property

## Abstract

Let $\{\infty^+, \infty^-\}$ be the two points above $\infty$ on the real hyperelliptic curve $H: y^2 = (x^2 - 1) \prod_{i=1}^{2g} (x - a_i)$. We show that the divisor $([\infty^+] - [\infty^-])$ is torsion in $\operatorname{Jac} J$ for a dense set of $(a_1, a_2, \ldots, a_{2g}) \in (-1, 1)^{2g}$. In fact, we prove by degeneration to a nodal $\mathbb{P}^1$ that an associated period map has derivative generically of full rank.