On the lifting problem in positive characteristic

Abstract

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.