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 $-K_S.C+p_g(C)-1$, 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 $\dim(V)=-K_S.C+p_g(C)-1$ 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 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

Moduli of surfaces in P3\mathbb{P}^3

The goal of this paper is to construct a compactification of the moduli space of degree $d \ge 5$ surfaces in $\mathbb{P}^3$, i.e. a parameter space whose interior points correspond to (equivalence classes of) smooth surfaces in $\mathbb{P}^3$ and whose boundary points correspond to degenerations of such surfaces. We study a more general problem and consider a divisor $D$ on a Fano variety $Z$ as a pair $(Z, D)$ satisfying certain properties. We find a modular compactification of such pairs and, in the case of $Z = \mathbb{P}^3$ and $D$ a surface, use their properties to classify the pairs on the boundary of the moduli space.Comment: Improved Theorem 1.1 to Fano varieties of any dimension and added results about the moduli space of pairs $(\mathbb{P}^n,D)$ for arbitrary $n$. Some typos fixed. 39 pages, comments welcome