  In this paper we prove a conjecture of Hershel Farkas that if a 4-dimensional
principally polarized abelian variety has a vanishing theta-null, and the
hessian of the theta function at the corresponding point of order two is
degenerate, the abelian variety is a Jacobian.
  We also discuss possible generalizations to higher genera, and an
interpretation of this condition as an infinitesimal version of Andreotti and
Mayer's local characterization of Jacobians by the dimension of the singular
locus of the theta divisor.
