Dimension counts for singular rational curves via semigroups

Subject: Mathematics  Combinatorics  14H20, 14H45, 14H51, 20Mxx  Mathematics  Commutative Algebra  Mathematics  Algebraic Geometry

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+ (i − g)g = ig − g ≥ (i + 1)i 2(i − 2) where P = V (x, y, z, w, x1, y1, z1, w1) is the usual Vandermonde determinant. It follows immediately that det(A0) 6= 0.
 S1∗ = {0, (1, 1), (2, 2)}, S2∗ = {0, (1, 1, 1)}. The analysis is similar to that of the preceding subcase. Recall that C2 is necessarily a triple point. Assuming that the branches x = (x : 1), y = (y : 1) and v = (v : 1), w = (w : 1), z = (z : 1) of C1 and C2, respectively, all lie in the same toric chart, the corresponding coefficient matrix A has a (maximal, rightmost) 7 × 7 submatrix A0 for which (up to a sign)  S∗ = {0, (1, 2), (2, 3)}, {0, (1, 1), (2, 2), (3, 3)}, or {0, (1, 1, 1), (2, 2, 1)}. These cases correspond to Cases 2, 3, and 5, respectively, of our classification of nonMT semigroups of genera 3 and 4 above. We discuss only the first possibility, as the arguments in the remaining two cases are completely analogous. Assume, then, that S∗ = {0, (1, 2), (2, 3)}, and that C1 is supported in (0 : · · · : 0 : 1) ∈ Pn, with branches whose preimages are 0 = (0 : 1) and ∞ = (1 : 0). Now say q = 1, so C2 is a simple cusp. Assume the preimage of the cusp is 1 = (1 : 1). The following conditions are operative, for all 0 ≤ i ≤ n − 1:
[6] A. Contiero, L. Feital, and R. V. Martins, Max Noether's theorem for integral curves, arXiv:1403.4167.
[7] A. Contiero and K.O. Sto¨hr, Upper bounds for the dimension of moduli spaces of curves with symmetric Weierstrass semigroups, J. London Math. Soc. 88 (2013), 580598.
[8] M. Coppens, Free linear systems on integral Gorenstein curves, J. Alg. 145 (1992), 209218.
[9] E. Cotterill, Rational curves of degree 11 on a general quintic threefold, Quart. J. Math. 63 (2012), no. 3, 539568.
[10] E. Cotterill, Rational curves of degree 16 on a general heptic fourfold, J. Pure Appl. Alg. 218 (2014), 121129.
[11] D. Cox, A. Kustin, C. Polini, and B. Ulrich, A study of singularities on rational curves via syzygies, Mem. Amer. Math. Soc. 222 (2013), no. 1045.
[12] D. Eisenbud and J. Harris, Divisors on general curves and cuspidal rational curves, Invent. Math. 74 (1983), 371418.
[13] D. Eisenbud, J. Koh, and M. Stillman, Determinantal equations for curves of high degree, Amer. J. Math. 110 (1988), no. 3, 513539.
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