Singularities of theta divisors and the geometry of A₅

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 [H] ∈ CH² (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 ⊂ A₅ has two components, both unirational, which we describe completely. 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 and the Iitaka dimension κ(A¯₅, N′¯₀) is equal to zero.