Exercise 111.52.7. Let $k$ be a field. Write $\mathbf{P}^3_ k = \text{Proj}(k[X_0, X_1, X_2, X_3])$. Let $C \subset \mathbf{P}^3_ k$ be a *type $(5, 6)$ complete intersection curve*. This means that there exist $F \in k[X_0, X_1, X_2, X_3]_5$ and $G \in k[X_0, X_1, X_2, X_3]_6$ such that

is a variety of dimension $1$. (Variety implies reduced and irreducible, but feel free to assume $C$ is nonsingular if you like.) Let $i : C \to \mathbf{P}^3_ k$ be the corresponding closed immersion. Being a complete intersection also implies that

is an exact sequence of sheaves. Please use these facts to:

compute $\chi (C, i^*\mathcal{O}_{\mathbf{P}^3_ k}(n))$ for any $n \in \mathbf{Z}$, and

compute the dimension of $H^1(C, \mathcal{O}_ C)$.

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