Non-degenerate curves with maximal Hartshorne-Rao module

Abstract

Abstract. Extending results for space curves we establish bounds for the cohomology of a non-degenerate curve in projective n-space. As a consequence, for any given n we determine all possible pairs (d, g) where d is the degree and g is the (arithmetic) genus of the curve. Furthermore, we show that curves attaining our bounds always exist and describe properties of these extremal curves. In particular, we determine the HartshorneRao module, the generic initial ideal and the graded Betti numbers of an extremal curve.

Even G-liaison classes of some unions of curves

“…The strategy of proof is similar to the one in [5,12], with two slight differences. Firstly, the general hyperplane section spans a linear space of dimension n − k, so we have h (1) n−k +1, and thus h 1 (I (1)) d −1.…”

“…In [12], Nagel proved a generalization of Hartshorne' s Restriction Theorem. He showed that if C ⊆ P n , n 4, and char(K) = 0, then the general hyperplane section of C is not contained in a line.…”

Non degenerate projective curves with very degenerate hyperplane section

“…From the bound on the genus computed in [9], Corollary 3.1, and from Proposition 3, we get that b ≥ 1. To show that h 0 (C, N C ) is equal to the lower bound computed in Corollary 2, we consider the compatibility condition, as computed in Remark 12, and so we have…”

“…In [9], Theorem 5.1, the author gives a presentation of the Hartshorne-Rao module of the curves with maximal cohomology and very degenerate general hyperplane section, and he proves the following theorem, that we translate to our notation: Theorem 8 ([9], Theorem 5.1). Let C ⊂ P n be a non degenerate curve with degree d ≥ 3 and genus g. Suppose that at least one of the three conditions…”

“…A generalization of Hartshorne's Restriction result was proved by U. Nagel in [9] for non degenerate locally Cohen-Macaulay curves C ⊂ P n , n ≥ 3, under the assumption char(K) = 0. In that paper, the author showed that if H is a general hyperplane with respect to C then C ∩ H is not contained in a line, but it can span a linear space of whatever dimension ≥ 2.…”