The Maximum Genus Problem for Locally Cohen-Macaulay Space Curves

摘要

Let P-MAX(d, s) denote the maximum arithmetic genus of a locally CohenMacaulay curve of degree d in P-3 that is not contained in a surface of degree s. A bound P(d, s) for P-MAX(d, s) has been proven by the first author in characteristic zero and then generalized in any characteristic by the third author. In this paper, we construct a large family C of primitive multiple lines and we conjecture that the generic element of C has good cohomological properties. From the conjecture it would follow that P(d, s) = P-MAX(d, s) for d = s and for every d = 2s - 1. With the aid of Macaulay2 we checked this holds for s = 120 by verifying our conjecture in the corresponding range.