The Maximal Rank Conjecture for sections of curves

摘要

Let C subset of P-r be a general curve of genus gembedded via a general linear series of degree d. The Maximal Rank Conjectureasserts that the restriction maps H-0(OPr(m)) -> H-0(O-C(m)) are of maximal rank; this determines the Hilbert function ofC. In this paper, we prove an analogous statement for the union of hyperplane sections of general curves. More specifically, if H. Pris a general hyperplane, and C-1, C-2,..., C-n are general curves, we show H-0(OH(m))-> H-0(O(C-1 boolean OR C-2 boolean OR center dot center dot center dot C-n)boolean AND H-(m)) is of maximal rank, except for some counterexamples when m = 2.