On the h-vector of a cohen-macaulay domain in positive characteristic

Here we study the Hilbert function of a Cohen-Macaulay homogeneous domain over an algebraically closed field of positive characteristic. The main tool (and an essential part of the main geometrical results) is the study of the Hilbert function of a general hyperplane section X?P ^r of an integral curve C?P ^r+1 , which is pathological in some sense. In §1, we study the case when Cis a strange curve, i.e., all tangent lines to Cat its simple points pass through a fixed point υ∈P ^r+1 . In §2, we give more refined results under the assumption that the Trisecant Lemma fails for C, i.e., any line spanned by two points of Ccontains one more point of C.