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 $g$?
 A: 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
A: 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.
A: 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.
