On the Severi problem in arbitrary characteristic

Karl Christ, Xiang He, Ilya Tyomkin

Published 2020-05-08Version 1

We show that Severi varieties parametrizing irreducible reduced planar curves of given degree and geometric genus are either empty or irreducible in any characteristic. As a consequence, we generalize Zariski's theorem to positive characteristic and show that a general reduced planar curve of given geometric genus is nodal. As a byproduct, we obtain the first proof of the irreducibility of the moduli space of smooth projective curves of given genus in positive characteristic, that does not involve a reduction to the characteristic zero case.