Supersingular curves of genus four in characteristic two

Unlike for abelian varieties, we know little about the curves with certain fixed invariants in positive characteristics. In this talk, we discuss how one can compute the dimension of the locus of genus-4 curves with an arbitrary eligible Newton polygon in characteristic 2; in particular, we show that the supersingular locus is 3-dimensional. The approach we present relies on the study of loci of curves with certain Ekedahl-Oort types lying inside the supersingular locus and can be modified to give some conclusions in some other characteristics as well.