## 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 + _{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) = -K_{S}.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 language | English |
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Pages (from-to) | 1783-1803 |

Number of pages | 21 |

Journal | Journal of the European Mathematical Society |

Volume | 15 |

Issue number | 5 |

DOIs | |

State | Published - 5 Aug 2013 |

## Keywords

- Curves on algebraic surfaces
- Severi varieties

## ASJC Scopus subject areas

- Mathematics (all)
- Applied Mathematics