A density result for real hyperelliptic curves

Abstract

Let {∞+,∞−} be the two points above ∞ on the real hyperelliptic curve H:y2=(x2−1)∏i=12g(x−ai). We show that the divisor ([∞+]−[∞−]) is torsion in JacJ for a dense set of (a1,a2,…,a2g)∈(−1,1)2g. In fact, we prove by degeneration to a nodal P1 that an associated period map has derivative generically of full rank.