On the lifting problem in positive characteristic

Given P nk with k algebraically closed field of characteristic p > 0, and X C Pnk integral variety of codimension 2 and degree d, let Y = X n H be the general hyperplane section of X. In this paper we study the problem of lifting, i.e. extending, a hypersurface in H of degree s containing Y to a hypersurface of same degree s in Pn containing X. For n = 3 and n = 4, in the case in which this extension does not exist we get upper bounds for d depending on s and p

On a theorem of Faltings on formal functions

In 1980, Faltings proved, by deep local algebra methods, a local result regarding formal functions which has the following global geometric fact as a consequence. Theorem: Let k be an algebraically closed field (of any characteristic). Let Y be a closed subvariety of a projective irreducible variety X defined over k. Assume that X \subseteq P^n, dim(X)=d>2 and Y is the intersection of X with r hyperplanes of P^n, with r \le d-1. Then, every formal rational function on X along Y can be (uniquely) extended to a rational function on X. Due to its importance, the aim of this paper is to provide two elementary global geometric proofs of this theorem.Comment: 9 page

ON THE SPECTRUM OF OCTAGON QUADRANGLE SYSTEMS OF ANY INDEX

An \emph{octagon quadrangle} is the graph consisting of a length $8$ cycle $(x_{1},x_{2},\dots,x_{8})$ and two chords, $\{x_{1},x_{4}\}$ and $\{x_{5},x_{8}\}$. An \emph{octagon quadrangle system} of order $v$ and index $\lambda$ is a pair $(X,\mathcal B)$, where $X$ is a finite set of $v$ vertices and $\mathcal B$ is a collection of octagon quadrangles (called blocks) which partition the edge set of $\lambda K_{v}$, with $X$ as vertex set. In this paper we determine completely the spectrum of octagon quadrangle systems for any index $\lambda$, with the only possible exception of $v=20$ for $\lambda=1$

Plane sections of space curves in positive characteristic

It is known that if C is a curve of degree d ≥ 6 in P^3 over an algebraically closed field of characteristic 0 such that its plane section is contained in an irreducible conic, then C lies on a quadric surface. We show under which conditions this result holds also in positive characteristic