Lemma 53.9.3. Let $Z \subset \mathbf{P}^2_ k$ be as in Lemma 53.9.1 and let $I(Z) = (F)$ for some $F \in k[T_0, T_1, T_2]$. Then $H^0(Z, \mathcal{O}_ Z) = k$ and the genus of $Z$ is $(d - 1)(d - 2)/2$ where $d = \deg (F)$.

**Proof.**
Let $S = k[T_0, T_1, T_2]$. There is an exact sequence of graded modules

Denote $i : Z \to \mathbf{P}^2_ k$ the given closed immersion. Applying the exact functor $\widetilde{\ }$ (Constructions, Lemma 27.8.4) we obtain

because $F$ generates the ideal of $Z$. Note that the cohomology groups of $\mathcal{O}_{\mathbf{P}^2_ k}(-d)$ and $\mathcal{O}_{\mathbf{P}^2_ k}$ are given in Cohomology of Schemes, Lemma 30.8.1. On the other hand, we have $H^ q(Z, \mathcal{O}_ Z) = H^ q(\mathbf{P}^2_ k, i_*\mathcal{O}_ Z)$ by Cohomology of Schemes, Lemma 30.2.4. Applying the long exact cohomology sequence we first obtain that

is an isomorphism and next that the boundary map

is an isomorphism. Since it is easy to see that the dimension of this is $(d - 1)(d - 2)/2$ the proof is finished. $\square$

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