Families of curves in positive characteristic

In this thesis, we study families of curves of genus g > 1 over fields of positive characteristic p > 0, focusing on invariants specific to this setting, such as the p-rank, Newton polygon, and Ekedahl-Oort type. A significant portion of the thesis is devoted to supersingular curves of genus g > 3, which are characterized by having the most “unusual” Newton polygon and exhibit many intriguing properties. In Chapter 3, we study the loci of curves of genera g = 4 and g = 5 in characteristic p = 2 in the corresponding moduli spaces of curves. In Chapter 4, we study Ekedahl-Oort types of stable curves and obtain an inductive bound on the dimension of the corresponding loci. In Chapter 5, we study the p-rank stratification of the loci of bielliptic curves in characteristic p > 2. In Chapter 6, we study supersingular curves of genus g = 4 in characteristic p > 2 with a non-trivial automorphism group. As a consequence, we prove Oort’s conjecture about the automorphism group of the generic point of every component of the supersingular locus of principally polarized abelian varieties in the case g = 4 and p > 2. Finally, in Chapter 7, we study the loci of double covers of curves and show the existence of smooth curves of genus g > 3 in characteristic p > 3 with certain unusual Newton polygons.