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The Stacks project

Exercise 111.56.7. In Situation 111.56.5 assume d = 5 and that the curve C = D is nonsingular. In the lectures we have shown that the genus of C is 6 and that the linear system K_ C is given by

L(K_ C) = \{ h\omega \mid h \in k[x, y],\ \deg (h) \leq 2\}

where \deg indicates total degree1. Let P_1, P_2, P_3, P_4, P_5 \in D be pairwise distinct points lying in the affine open X_0 \not= 0. We denote \sum P_ i = P_1 + P_2 + P_3 + P_4 + P_5 the corresponding divisor of C.

  1. Describe L(K_ C - \sum P_ i) in terms of polynomials.

  2. What are the possibilities for l(\sum P_ i)?

[1] We get \leq 2 because d - 3 = 5 - 3 = 2.

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