Geodesic
On Zariski's theorem in positive characteristic
Journal of the European Mathematical Society, Tome 15 (2013) no. 5, pp. 1783-1803

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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 −KS​.C+pg​(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)=−KS​.C+pg​(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.
DOI : 10.4171/jems/403
Classification : 14-XX, 00-XX
Keywords: Curves on algebraic surfaces, Severi varieties 
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     title = {On {Zariski's} theorem in positive characteristic},
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Ilya Tyomkin. On Zariski's theorem in positive characteristic. Journal of the European Mathematical Society, Tome 15 (2013) no. 5, pp. 1783-1803. doi: 10.4171/jems/403

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