  We generalize the notion of quasielliptic curves, which have infinitesimal
symmetries and exist only in characteristic two and three, to a remarkable
hierarchy of regular curves having infinitesimal symmetries, defined in all
characteristics and having higher genera. This relies on the study of certain
infinitesimal group schemes acting on the affine line and certain
compactifications. The group schemes are defined in terms of invertible
additive polynomials over rings with nilpotent elements, and the
compactification is constructed with the theory of numerical semigroups. The
existence of regular twisted forms relies on Brion's recent theory of
equivariant normalization. Furthermore, extending results of Serre from the
realm of group cohomology, we describe non-abelian cohomology for semidirect
products, to compute in special cases the collection of all twisted forms.
