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 sectionX⊂Prof an integral curveC⊂Pr+1, which is pathological in some sense. In §1, we study the case whenCis a strange curve, i.e., all tangent lines toCat its simple points pass through a fixed point υ∈Pr+1. In §2, we give more refined results under the assumption that the Trisecant Lemma fails forC, i.e., any line spanned by two points ofCcontains one more point ofC.