A density result for real hyperelliptic curves

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