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

where $\deg $ indicates total degree^{1}. 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 $C$.

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

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

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