Let $C$ be a smooth curve of genus$\geq3$ over $\mathbb{C}$, so there are $2^{g-1}(2^g-1)$ odd theta characteristics and $2^{g-1}(2^g+1)$ even theta characteristics. Do we know how many of them has $h^0=0$, $h^0=1$, $h^0=2$...? If $h^0\geq2$, are they base-point free?


In particular, I would like to know the following.
If $g=3$, is there a non-hyperelliptic curve with some $h^0\geq2$ base-point free theta characteristic?
The same question for hyperelliptic curves for general $g$?


References are appreciated! Thanks!



Theta characteristics of genus$\geq3$ curve
A lot is known about this problem. 


Firt of all, the answer to your first question is **no**. In fact, we can prove the following easy result:



> 
> **Proposition.** A curve $C$ of genus $3$ is hyperelliptic if and only if it contains a theta characteristic $L$ such that $h^0(L) \geq 2$.
> 
> 
> *Proof.* Assume that $C$ is not hyperelliptic. Then on the canonical model of $C$, which is a plane quartic, an effective theta characteristic is cut out by a contact hyperplane, i.e. a bitangent line. Since $h^0(L) \geq 2$, we must have infinitely many contact hyperplanes, absurd because any plane quartic contains precisely $28$ bitangent lines. 
> 
> 
> Conversely, if $C$ is hyperelliptic then $|K\_C|$ is composed with the unique $g\_2^1$, and since $\deg K\_C=4$ such a $g^1\_2$ is a theta characteristic $L$ with $h^0(L)=2$. $\square$
> 
> 
> 


For the general case, let me just state a single result, referring to [1](http://www.ams.org/journals/tran/1987-302-01/S0002-9947-1987-0887499-X/) and the references given therein for a more complete treatment.



> 
> **Theorem.** Denote by $\mathscr{M}^r\_g$ the sublocus of $\mathscr{M}\_g$ consisting of curves having a theta characteristic $L$ such that $$h^0(L) \geq r+1 \quad \textrm{and} \quad h^0(L) \equiv r+1 \, (\textrm{mod } 2).$$
>  Then $\mathscr{M}\_g^1$ (resp. $\mathscr{M}\_g^2$) has pure codimension $1$ (resp. $3$) in $\mathscr{M}\_{g}$ if $g \geq 3$ (resp. $g \geq 5$) and a generic point of any of its components is a curve which has only one theta-characteristic $L$ with $h^0(L)=2$ (resp. $h^0(L) =3$ if $g \geq 6$). 
> 
> 
> Moreover if $g \geq 3$ such a $L$ is not composed with an involution (resp. if $g \geq 6$ such a $L$ has no fixed points). 
> 
> 
> 


**References.**


[1](http://www.ams.org/journals/tran/1987-302-01/S0002-9947-1987-0887499-X/) Montserrat Teixidor I Bigas: [Half canonical series on algebraic curves](http://www.ams.org/journals/tran/1987-302-01/S0002-9947-1987-0887499-X/), *Trans. Amer. Math. Soc.* **302** (1987), 99-115.


