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Suppose I am given a Riemann surface $\Sigma_g$ of genus $g$. What is known about the sufficient and necessary conditions needed on $\Sigma_g$ to have an embedding into $\mathbb P^2$?

If $\mathcal T_g$ is the Teichmuller space of genus $g$ Riemann surfaces, is there any way to intrinsically describe the subspace of plane curves?

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    $\begingroup$ If $g > 0$, then an embedding is not surjective (because $\Sigma_g \not\cong \mathbb{P}^1$), so there is a point $p \in \mathbb{P}^1$ not in the image, so $\Sigma_g$ embeds into $\mathbb{P}^1\setminus\{p\} \cong \mathbb{C}$ which is impossible as $\mathbb{C}$ does not have compact submanifolds other than finite sets of points. $\endgroup$
    – Michael Albanese
    Commented Mar 15, 2017 at 20:41
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    $\begingroup$ Sorry, there was a typo. I meant $\mathbb P^2$, not $\mathbb P^1$. Sorry about that. $\endgroup$
    – Harry Reed
    Commented Mar 15, 2017 at 20:57
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    $\begingroup$ What kind of data do you have for your mystery surface? $\endgroup$
    – Igor Rivin
    Commented Mar 15, 2017 at 21:01
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    $\begingroup$ I would be surprised if there were any sufficient conditions known that were a significant departure from an immediate tautology. Call me sceptical. $\endgroup$
    – Ryan Budney
    Commented Mar 15, 2017 at 21:02
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    $\begingroup$ By the genus formula for smooth plane curves of degree $d$, unless $g=(d-1)(d-2)/2$ for some integer $d\ge 1$ the answer is negative. I don't know what happens for the "exceptional" values. $\endgroup$
    – Abdelmalek Abdesselam
    Commented Mar 15, 2017 at 21:19

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If I am not mistaken, for the exceptional values of the genus, the locus of degree $d$ plane curves has (complex) dimension $$ \frac{(d+2)(d+1)}{2}-9 $$ sitting inside a moduli space of dimension $$ \frac{3(d-1)(d-2)}{2}-3 $$ so it becomes relatively small as $d$ increases.

This article by Landesman seems to be pertinent for your question.

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  • $\begingroup$ Thanks for the reference! I am completely unfamiliar with stacks, so I guess it's time to study! $\endgroup$
    – Harry Reed
    Commented Mar 16, 2017 at 3:20
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    $\begingroup$ Given a curve of genus $ g = \frac{(d-1)(d-2)}{2} $ asking "is this a plane curve' is very hard afaik. I'll add two comments in case they are helpful to your situation. Firstly, that the curve poses a complete $g^2_d$ is obviously necessary, but isn't sufficient. A simple example would be a curve of genus 5 that is trigonal. The 'complementary' linear system is a $g^2_5$ . Secondly, I think that Lazarsfeld (with co-authors ?) has shown that at least 'usually ' a $g^2_d$ on a curve is unique. hth $\endgroup$
    – meh
    Commented Mar 16, 2017 at 14:24
  • $\begingroup$ I'm not knowledgeable about Teichmuller theory, so I won't attempt an answer to your question. However, there is a paper by Marc Coppens, " On G. Martens' Characterization of Plane Curves" in which he proves that if X has d+1 mutually independent $g^1_d$'s, then X is isomorphic to a plane curve of degree d+1. The paper is behind a pay wall, so I haven't been able to read the details. The paper claims this bound is sharp. I don't know what this means from a Teichmuller point of view. From an algebraic geometry view, it seems interesting to me. $\endgroup$
    – meh
    Commented Mar 23, 2017 at 19:04

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