In the current paper we show that the dimension of a family of irreducible reduced curves in a given ample linear system on a toric surface over an algebraically closed field is bounded from above by , where 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 does not imply the nodality of even if belongs to the smooth locus of , 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.
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Ilya Tyomkin, On Zariski's theorem in positive characteristic. J. Eur. Math. Soc. 15 (2013), no. 5, pp. 1783–1803DOI 10.4171/JEMS/403