Question

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 $g$?

Answer 1

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.

Answer 2

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: http://www.math.lsa.umich.edu/~idolga/topics.pdf