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...