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

On the Schottky problem for genus five Jacobians with a vanishing theta null