Dimension counts for singular rational curves via semigroups

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Cotterill, Ethan ; Feital, Lia ; Martins, Renato Vidal (2015)
  • Subject: Mathematics - Combinatorics | 14H20, 14H45, 14H51, 20Mxx | Mathematics - Commutative Algebra | Mathematics - Algebraic Geometry

We study singular rational curves in projective space, deducing conditions on their parametrizations from the value semigroups $\sss$ of their singularities. In particular, we prove that a natural heuristic for the codimension of the space of nondegenerate rational curves of arithmetic genus $g>0$ and degree $d$ in $\mb{P}^n$, viewed as a subspace of all degree-$d$ rational curves in $\mb{P}^n$, holds whenever $g$ is small.
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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 non-MT 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:

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