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
\[ C = \text{Proj}(k[X_0, X_1, X_2, X_3]/(F, G)) \]
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
\[ \xymatrix{ 0 \ar[r] & \mathcal{O}_{\mathbf{P}^3_ k}(-11) \ar[r]^-{ \left( \begin{matrix} -G
\\ F
\end{matrix} \right) } & \mathcal{O}_{\mathbf{P}^3_ k}(-5) \oplus \mathcal{O}_{\mathbf{P}^3_ k}(-6) \ar[r]^-{(F, G)} & \mathcal{O}_{\mathbf{P}^3_ k} \ar[r] & i_*\mathcal{O}_ C \ar[r] & 0 } \]
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 $H^1(C, \mathcal{O}_ C)$.
Comments (0)