  We study the degree of the Gauss map of the theta divisor of principally
polarised complex abelian varieties. We use this to obtain a bound on the
multiplicity of the theta divisor along irreducible components of its singular
locus, and apply this bound in examples, and to understand the local structure
of isolated singular points. We further define a stratification of the moduli
space of ppav's by the degree of the Gauss map. In dimension four, we show that
this stratification gives a weak solution of the Schottky problem, and we
conjecture that this is true in any dimension.
