A Density Result for Real Hyperelliptic Curves

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