Generic initial ideals of singular curves in graded lexicographic order

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⩾5d⩾5 with the arithmetic genus ρa(C)ρa(C) in PrPr where r⩾3r⩾3. If M(IC)M(IC) is the regularity of the lexicographic generic initial ideal of ICIC in a polynomial ring k[x0,…,xr]k[x0,…,xr] then we prove that M(IC)M(IC) is 1+(d−12)−ρa(C) which is obtained from the monomialxr−3xr−1(d−12)−ρa(C), provided that dimTanp(C)=2 for every singular point p∈Cp∈C. This number is equal to one plus the number of secant lines through the center of general projection into P2P2. 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 P3P3. 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]).