Exercise 111.54.3. Let $k$ be a field. Let $F \in k[X_0, X_1, X_2]$ be a homogeneous form of degree $d$. Assume that $C = V_{+}(F) \subset \mathbf{P}^2_ k$ is a smooth curve over $k$. Denote $i : C \to \mathbf{P}^2_ k$ the corresponding closed immersion.

Show that there is a short exact sequence

\[ 0 \to \mathcal{O}_{\mathbf{P}^2_ k}(-d) \to \mathcal{O}_{\mathbf{P}^2_ k} \to i_*\mathcal{O}_ C \to 0 \]of coherent sheaves on $\mathbf{P}^2_ k$: tell me what the maps are and briefly why it is exact.

Conclude that $H^0(C, \mathcal{O}_ C) = k$.

Compute the genus of $C$.

Assume now that $P = (0 : 0 : 1)$ is not on $C$. Prove that $\pi : C \to \mathbf{P}^1_ k$ given by $(a_0 : a_1 : a_2) \mapsto (a_0 : a_1)$ has degree $d$.

Assume $k$ is algebraically closed, assume all ramification indices (the “$e_ i$”) are $1$ or $2$, and assume the characteristic of $k$ is not equal to $2$. How many ramification points does $\pi : C \to \mathbf{P}^1_ k$ have?

In terms of $F$, what do you think is a set of equations of the set of ramification points of $\pi $?

Can you guess $K_ C$?

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