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Let $C$ be a complex smooth projective curve of genus $g$ and let $N$ be the number of effective theta-characteristics of $C$, or equivalently, the number of points of order two on the theta divisor of $J(C)$.

It is known that, if $C$ is generic, $N$ is the number of the odd theta characteristics. Mumford proves that on a principally polarized abelian variety the theta divisor cannot contain all the points of order two. It follows that $N<2^{2g}$.

Given an arbitrary curve $C$, is it known a upper bound for $N$ depending on $g$?

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3 Answers 3

Well, there's a lower bound as odd theta characteristics on a canonical curve are effective, so there are at least $2^{g-1}(2^g -1)$ of them. Even thetas are trickier.

Please consult Dolgachev's book:

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Yes, I mean upper bound. Thank you. Now I've modified the question and it is more precise. I take a look to your link. – V M Sep 29 '10 at 14:38
Probably of especial interest to you would be the Scorza correspondence if you don't already know about it. – stankewicz Sep 29 '10 at 14:41
I didn't know about it. I'm not sure I understand your answer. Do you mean that Scorza correspondence gives an estimate on the maximal number of effective theta characteristics? – V M Oct 2 '10 at 14:46
Well, not directly but you need a non-effective theta characteristic to define such a correspondence and a correspondence on $C\times C$ of the type detailed in Dolgachev's book gives a non-effective theta. So while I don't know of a non-trivial upper bound, there's at least a foothold you can start from if you wanted to prove something. – stankewicz Oct 2 '10 at 15:35

In genus 4 it seems the maximum number of vanishing even theta nulls is 10, which in fact occurs on a unique 4 dimensional principally polarized abelian variety. A bound may be obtained by considering the effect on the degree of the Gauss map of the theta divisor.

You may consult the paper of Robert Varley:

oops these are perhaps the isolated singularities on theta. I have not checked but the non isolated case of hyperelliptic jacobians may be different. Lets see, a h.e jacobian of genus 4 occurs as a double cover of P^1 branched at 10 points, so there are I guess, gosh again it seems there are 10 of them, i.e. the hyperelliptic line bundle plus one of the 10 ramification points.

The ranks of the double points are all 3 in this case, and are all 4 in the previous isolated case.

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Since the odd theta characteristics are always effective, one might equivalently ask how many even theta characteristics are effective. They are called vanishing theta characteristics.

If $C$ is a hyperelliptic curve of genus $g$, then there are $ c_g = \frac{1}{2} \left( \begin{array}{c} 2g+2 \\ g+1 \end{array} \right) $ even theta characteristics that do not vanish$^1$. Which means, the number $N$, defined in your question, for hyperelliptic curves of genus $g$ is $2^{2g} - c_g$.

This at least provides a lower bound for an upper bound for $N$.

Arnaud Beauville in Vanishing thetanulls on curves with involution looks further. However, he concludes his paper by saying that what you are asking is open for non-hyperelliptic curves even in $g \ge 6$.

It could be possible that the maximum $N$ is attained on curves with involution, or even on hyperelliptic curves. If you could show that, you would be able to answer your question using these results.

[1] See the proof of Lemma 5.2.2 in Dolgachev's Classical Algebraic Geometry.

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