On Zariski's theorem in positive characteristic

TL;DR

This paper extends Zariski's theorem to positive characteristic, providing bounds on the dimension of families of curves on toric surfaces and revealing new phenomena unique to positive characteristic settings.

Contribution

It generalizes Zariski's theorem to positive characteristic and constructs examples of reducible Severi varieties on weighted projective planes.

Findings

Dimension bound for families of curves on toric surfaces

Equality does not imply nodality in positive characteristic

Construction of reducible Severi varieties in positive characteristic

Abstract

In the current paper we show that the dimension of a family $V$ of irreducible reduced curves in a given ample linear system on a toric surface $S$ over an algebraically closed field is bounded from above by $-K_S.C+p_g(C)-1$, where $C$ denotes a general curve in the family. This result generalizes a famous theorem of Zariski to the case of positive characteristic. We also explore new phenomena that occur in positive characteristic: We show that the equality $\dim(V)=-K_S.C+p_g(C)-1$ does not imply the nodality of $C$ even if $C$ belongs to the smooth locus of $S$, and construct reducible Severi varieties on weighted projective planes in positive characteristic, parameterizing irreducible reduced curves of given geometric genus in a given ample linear system.