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 $g\geq 8$ 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 $g\geq 6$, the hyperelliptic locus ${{\mathcal{M}}}^1_{g,2}$ is the only $2g-1$-dimensional sub-locus ${{\mathcal{Y}}}$ of the moduli space ${{\mathcal{M}}}_g$ of curves of genus $g$, such that for the general element $[C]\in{{\mathcal{Y}}}$, its Jacobian $J(C)$ is dominated by a hyperelliptic Jacobian of genus $g^{\prime}\geq g$.