On Zariski's theorem in positive characteristic

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.