JournalsjemsVol. 15, No. 5pp. 1783–1803

On Zariski's theorem in positive characteristic

  • Ilya Tyomkin

    Ben Gurion University of the Negev, Beer Sheva, Israel
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In the current paper we show that the dimension of a family VV of irreducible reduced curves in a given ample linear system on a toric surface SS over an algebraically closed field is bounded from above by KS.C+pg(C)1-K_S.C+p_g(C)-1, where CC 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\dim(V)=-K_S.C+p_g(C)-1 does not imply the nodality of CC even if CC belongs to the smooth locus of SS, 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