# On Zariski's theorem in positive characteristic

### Ilya Tyomkin

Ben Gurion University of the Negev, Beer Sheva, Israel

## 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 very ample linear system.

## Cite this article

Ilya Tyomkin, On Zariski's theorem in positive characteristic. J. Eur. Math. Soc. 15 (2013), no. 5, pp. 1783–1803

DOI 10.4171/JEMS/403