Trigonal Deformations of Rank One and Jacobians

We study the infinitesimal deformations of a trigonal curve that preserve the trigonal series and such that the associate infinitesimal variation of Hodge structure is of rank 1. We show that if g≥8 or g=6,7 and the curve is Maroni general, this locus is zero dimensional. Moreover, we complete the result [10, Theorem 1.6]. We show in fact that if g≥6⁠, the hyperelliptic locus is the only 2g−1-dimensional sub-locus of the moduli space of curves of genus g⁠, such that for the general element its Jacobian is dominated by a hyperelliptic Jacobian of genus g′≥g⁠.