Generic initial ideals of singular curves in graded lexicographic order

Abstract

In this paper, we are interested in the generic initial ideals of singular projective curves with respect to the graded lexicographic order. Let C be a singular irreducible projective curve of degree d >= 5 with the arithmetic genus rho(a)(C) in P-r where r >= 3. If M(I-C) is the regularity of the lexicographic generic initial ideal of I-C in a polynomial ring k[chi(0,) . . . . chi(r)] then we prove that M(I-C) is 1 + ((d-1)(2)) - rho(a)(C) which is obtained from the monomial [GRAPHICS] provided that dim Tan(p)(C) = 2 for every singular point p is an element of C. This number is equal to one plus the number of secant lines through the center of general projection into P-2. Our result generalizes the work of J. Ahn (2008) [1] for smooth projective curves and that of A. Conca and J. Sidman (2005) [9] for smooth complete intersection curves in P-3. The case of singular curves was motivated by A. Conca and J. Sidman (2005) [9, Example 4.3]. We also provide some illuminating examples of our results via calculations done with Macaulay 2 and singular (Decker et al., 2011 [10], Grayson and Stillman [16]). (C) 2012 Elsevier Inc. All rights reserved.