A Density Result for Real Hyperelliptic Curves

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 $\mathrm{Jac}J$ for a dense set of $(a_1,a_2,\dots ,a_{2g})\in (-1,1{)}^{2g}$. In fact, we prove by degeneration to a nodal ${\mathbb{P}}^1$ that an associated period map has derivative generically of full rank.

9 pages; added "Prior Work" section