Proposition 53.10.4 (Characterization of the projective line). Let $k$ be a field. Let $X$ be a proper curve over $k$. The following are equivalent

$X \cong \mathbf{P}^1_ k$,

$X$ is smooth and geometrically irreducible over $k$, $X$ has genus $0$, and $X$ has an invertible module of odd degree,

$X$ is geometrically integral over $k$, $X$ has genus $0$, $X$ is Gorenstein, and $X$ has an invertible sheaf of odd degree,

$H^0(X, \mathcal{O}_ X) = k$, $X$ has genus $0$, $X$ is Gorenstein, and $X$ has an invertible sheaf of odd degree,

$X$ is geometrically integral over $k$, $X$ has genus $0$, and $X$ has an invertible $\mathcal{O}_ X$-module of degree $1$,

$H^0(X, \mathcal{O}_ X) = k$, $X$ has genus $0$, and $X$ has an invertible $\mathcal{O}_ X$-module of degree $1$,

$H^1(X, \mathcal{O}_ X) = 0$ and $X$ has an invertible $\mathcal{O}_ X$-module of degree $1$,

$H^1(X, \mathcal{O}_ X) = 0$ and $X$ has closed points $x_1, \ldots , x_ n$ such that $\mathcal{O}_{X, x_ i}$ is normal and $\gcd ([\kappa (x_ i) : k]) = 1$, and

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