  We study the codimension two locus H in A_g consisting of principally
polarized abelian varieties whose theta divisor has a singularity that is not
an ordinary double point. We compute the class of H in A_g for every g. For
g=4, this turns out to be the locus of Jacobians with a vanishing theta-null.
For g=5, via the Prym map we show that H in A_5 has two components, both
unirational, which we completely describe. This gives a geometric
classification of 5-dimensional ppav whose theta-divisor has a quadratic
singularity of non-maximal rank.
  We then determine the slope of the effective cone of A_5 and show that the
component N_0' of the Andreotti-Mayer divisor has minimal slope 54/7.
Furthermore, the Iitaka dimension of the linear system corresponding to N_0' is
submaximal.
