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
βKSβ.C+pgβ(C)β1, 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)=βKSβ.C+pgβ(C)β1 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.Comment: 19 pages. Several typos have been fixed, and a couple of examples and
pictures have been added. To appear in JEM