Correspondence between bitangents of a quartic and odd theta characteristics

Question

Let $C$ be a Riemann surface of genus $g=3$. I can't understand why the following statements are true:

If $C$ is not hyperelliptic, then the canonical series $|K|$ embeds $C$ as a smooth quartic in $\mathbb{P}^2$
The odd theta characteristics of $C$ correspond to the bitangent lines to this quartic.

How can these facts be explained?

How do you know about these facts? And what is an odd theta characteristic? — Georges Elencwajg, Jun 28 '15 at 13:07
They are stated in a lot of articles and books (e.g. http://www.math.harvard.edu/~gross/preprints/theta.pdf , p.15). A theta characteristic is a divisor class which is half the canonical class. An odd one is one, say $L$, such that $\mathrm{dim}H^0(C,L)$ is odd. — Davide, Jun 28 '15 at 13:32
The first bit is probably in Hartshorne's section IV.5 on canonical embeddings. — Hoot, Jun 28 '15 at 22:48

score 2 · Answer 1 · answered Jun 28 '15 at 14:42

Since for a canonical divisor $K$ the linear system $|K|$ has no base point as soon as $g\geq1$ ( this results from Riemann-Roch), the canonical map $i_K:C\to \mathbb P^2$ is an everywhere defined morphism.
The only way it could fail to be an embedding is if for some $p,q\in C$ (not necessarily distinct) we had $\dim_\mathbb C \Gamma(C,\mathcal O(p+q))=2$ : this follows from Riemann-Roch too.
But this is equivalent to $C$ being hyperelliptic.
In the other cases $i_K$ is an embedding and the degree of its (smooth) image is the degree of the divisor $K$, namely $2g-2=2.3-2=4$.
You can find supplementary details on page 247 of Griffiths-Harris's Principles of Algebraic Geometry.

Thank you, this answers my first question. Any idea about the correspondence between bitangents and odd theta characteristics? — Davide, Jun 28 '15 at 22:28
The canonical class is the pullback of the hyperplane section by adjunction. So every bitangent will give an effective divisor whose double is the canonical class. — Lev Borisov, Jul 07 '15 at 00:26

Correspondence between bitangents of a quartic and odd theta characteristics

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