  Inside the moduli space of curves of genus three with one marked point, we
consider the locus of hyperelliptic curves with a marked Weierstrass point, and
the locus of non-hyperelliptic curves with a marked hyperflex. These loci have
codimension two. We compute the classes of their closures in the moduli space
of stable curves of genus three with one marked point. Similarly, we compute
the class of the closure of the locus of curves of genus four with an even
theta characteristic vanishing with order three at a certain point. These loci
naturally arise in the study of minimal strata of Abelian differentials.
