  We give a solution to the weak Schottky problem for genus five Jacobians with
a vanishing theta null, answering a question of Grushevsky and Salvati Manni.
More precisely, we show that if a principally polarized abelian variety of
dimension five has a vanishing theta null with a quadric tangent cone of rank
at most three, then it is in the Jacobian locus, up to extra irreducible
components. We employ a degeneration argument, together with a study of the
ramification loci for the Gauss map of a theta divisor.
