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?

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    $\begingroup$ One observation is: $X\backslash\{x\}$ is always affine. This is because $nx$ is very ample for some $n$ sufficiently big, and so defines an embedding $X\hookrightarrow\mathbb{P}^r$ for $r=h^0(\mathcal{O}_X(nx))$. Now with this embedding, there is a hyperplane $H\subseteq\mathbb{P}^r$ such that $H\cap X=\{x\}$ (set theoretically), and so $X\backslash\{x\}\subseteq\mathbb{P}^r\backslash H\simeq\mathbb{A}^r$. Don't know if this helps at all. $\endgroup$ – rfauffar Oct 6 '13 at 0:47
  • $\begingroup$ @RobertAuffarth Thanks! So we need to choose points so that the remaining is an affine plane curve... Maybe using something that IF we remove such a point,$x_0$ $\omega_{X \setminus x_0} $ should be isomorphic to $\mathcal{O}_C$. $\endgroup$ – Tedar Oct 6 '13 at 10:24
  • $\begingroup$ I have a hunch that this should connect to the Jacobian and the fact that over an algebraically closed field, for X a smooth, connected projective curve of genus g, $Pic^0(X)[n] \cong (\mathbb{Z}/nZ)^{2g}$. $\endgroup$ – Tedar Oct 6 '13 at 13:53
  • $\begingroup$ Dear @rfauffar, your comment is a great one but the embedding you mention is into $\mathbb P^{r-1}$. $\endgroup$ – Georges Elencwajg Nov 13 '14 at 10:05
  • $\begingroup$ Dear @GeorgesElencwajg, thank you for your comment; you are absolutely right. Unfortunately I can't edit the comment. $\endgroup$ – rfauffar Nov 13 '14 at 13:04

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.

  • $\begingroup$ @GeorgesElencwajg yes, that was a typo due to being in a rush. Now the propf should make sense. Thanks for your comment! $\endgroup$ – Dedalus Nov 13 '14 at 18:51
  • $\begingroup$ @GeorgesElencwajg This is in part a) of the mentioned exercise, pg. 302 of Liu's book. $\endgroup$ – Dedalus Nov 13 '14 at 20:09
  • $\begingroup$ @GeorgesElencwajg I am sorry if I didn't mention that I changed to plane affine, I thought that you would reread it to see if it made sense now. I have no intent of bad etiquette. I have not checked exercise a) , but since the question explicitly assumes it I didn't do it. I might try to do it if I have the time and know how to do it. Lastly, I did not mean to imply that your statement was incorrect, rather give a reference for the assertion I made. $\endgroup$ – Dedalus Nov 13 '14 at 23:07
  • $\begingroup$ @Dear Dedalus: ok, misunderstanding cleared. I have removed my comments. $\endgroup$ – Georges Elencwajg Nov 13 '14 at 23:24
  • $\begingroup$ @Dedalus Could you explain how to deduce $K~(2g-1)x$ from $w_{C/k}\cong O_C$? Thanks! $\endgroup$ – Sssss Nov 5 '19 at 13:40

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