TITLE: Curve minus finite number of points affine
QUESTION [9 upvotes]: I am doing another exercise from Liu.
let X be a smooth geometrically connected projective curve over a field k of genus $g \geq 2$ Show that there exist at most $(2g-2)^{2g}$ points $x \in X(k)$ such that $X \setminus x$ is an affine plane curve.
In the first exercise, one showed that $\omega_{C/k} \cong \mathcal{O_C}$ if C is an affine plane curve, i.e a curve iomorphic to a closed subcheme of an open subscheme of $\mathbb{A}^2_k$. My thinking was that maybe we should use that the degree of the canonical divisor on X is $2g-2$, and then... I am not sure. Any hint?

REPLY [2 votes]: Let us assume $k$ is algebraically closed. I believe the method can be extended for k not algebraically closed, I will try to write a follow-up later. Let $X$ be your curve and note that if $x$ is a point such that $C= X \setminus x$ is an affine plane curve, then we have that $\omega_{C/k} \cong \mathcal{O}_C$ so that $K \sim (2g-2)x$ for $K$ the canonical divisor. So let us fix a point $x \in X$ such that $X \setminus x$ is an affine plane curve. Then we have that for any other point $x'$ such that $X \setminus x'$ is an affine plane curve, that  $(2g-2)(x-x') \equiv 0.$ Thus, $x-x'$ is of order $(2g-2)$. Now, we have that $Pic^0(X)(2g-2) \cong (\mathbb{Z}/(2g-2))^{2g}$ thus, there can be at most $2g-2$ such points.
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