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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View Exercise 111.56.7 as pdf