On Zariski's theorem in positive characteristic

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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.

Original languageEnglish
Pages (from-to)1783-1803
Number of pages21
JournalJournal of the European Mathematical Society
Issue number5
StatePublished - 5 Aug 2013


  • Curves on algebraic surfaces
  • Severi varieties

ASJC Scopus subject areas

  • Mathematics (all)
  • Applied Mathematics


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