# Effective theta characteristics

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 $g$?

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.

• 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, 2010 at 14:38
• Probably of especial interest to you would be the Scorza correspondence if you don't already know about it. Sep 29, 2010 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, 2010 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. Oct 2, 2010 at 15:35

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.

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

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: http://www.jstor.org/pss/2374519

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.