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.
|Original language||English GB|
|State||Published - 2020|
- 14D20, 14H10, 14H50, 14T05