Lemma 53.10.3. Let X be a proper curve over a field k with H^0(X, \mathcal{O}_ X) = k. If X is Gorenstein and has genus 0, then X is isomorphic to a plane curve of degree 2.
Proof. Consider the invertible sheaf \mathcal{L} = \omega _ X^{\otimes -1} where \omega _ X is as in Lemma 53.4.1. Then \deg (\omega _ X) = -2 by Lemma 53.8.3 and hence \deg (\mathcal{L}) = 2. By Lemma 53.10.2 we conclude that choosing a basis s_0, s_1, s_2 of the k-vector space of global sections of \mathcal{L} we obtain a closed immersion
\varphi _{(\mathcal{L}, (s_0, s_1, s_2))} : X \longrightarrow \mathbf{P}^2_ k
Thus X is a plane curve of some degree d. Let F \in k[T_0, T_1, T_2]_ d be its equation (Lemma 53.9.1). Because the genus of X is 0 we see that d is 1 or 2 (Lemma 53.9.3). Observe that F restricts to the zero section on \varphi (X) and hence F(s_0, s_1, s_2) is the zero section of \mathcal{L}^{\otimes 2}. Because s_0, s_1, s_2 are linearly independent we see that F cannot be linear, i.e., d = \deg (F) \geq 2. Thus d = 2 and the proof is complete. \square
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