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 $-K_S.C+p_g\left(C\right)-1$, 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 $\mathrm{𝚍𝚒𝚖}\left(V\right)=-K_S.C+p_g\left(C\right)-1$ does not imply the nodality of $C$ even if $C$ belongs to the...

Let $C$ be an algebraic projective smooth and trigonal curve of genus $g\ge 5$. In this paper we define, in a way equivalent to that followed by A. Maroni in [1], an integer $m$, called the species of $C$, which is a birational invariant having the property that $0\le m\le \frac{g+2}{3}$ and $m\mathrm{—}g\equiv 0$ mod(2). In section 1 we prove that for every $g(\ge 5)$ and every $m$, as before, there are trigonal curves of genus $g$ and species $m$. In section 2 we define the space ${ℳ}_{g,3;m}^1$ of moduli of trigonal curves of genus $g$ and species $m$. We note that ${ℳ}_{g,3;m}^1$ is irreducible...