030625 Dusan Dragutinovic

While the properties of families of abelian varieties with respect to the Newton polygon stratification in characteristic p > 0 are generally well understood, relatively little is known about families of smooth curves of genus g as soon as g > 3. Supersingular curves are those with the most unusual Newton polygon. Their existence for g = 4 and any prime p > 0 was only recently established by Kudo-Harashita-Senda and independently by Pries. In this talk, we present some results on the dimensions of the loci of supersingular curves of genus g = 4, as well as on their automorphism groups. As an outcome, our results confirm Oort's conjecture about the generic automorphism group of the supersingular locus of principally polarized abelian varieties for g = 4 and p > 2.