# On Zariski's theorem in positive characteristic Academic Article

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### 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.

### publication date

• January 1, 2013