Initial ideals of weighted forms and the genus of locally Cohen-Macaulay curves
arXiv:2501.00809 · doi:10.1007/s00209-026-03946-4
Abstract
Let C be a locally Cohen-Macaulay curve in complex projective 3-space. The maximum genus problem predicts the largest possible arithmetic genus g(d,s) that C can achieve assuming that it has degree d and does not lie on surfaces of degree less than s. In this paper, we prove that this prediction is correct when d=s or d is at least 2s-1. We obtain this result by proving another conjecture, by Beorchia, Lella, and the second author, about initial ideals associated to certain homogeneous forms in a non-standard graded polynomial ring.
49 pages, 6 figures. Comments are welcome!