Theta characteristics of genus$\geq3$ curve

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!

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 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 Montserrat Teixidor I Bigas: Half canonical series on algebraic curves, Trans. Amer. Math. Soc. 302 (1987), 99-115.

• I think you mean h^0(L) congruent to r+1 (mod 2)? It is somewhat confusing in the paper cited, but she seems to mean projective dimension congruent to r. Commented Jun 25, 2015 at 18:29

As an alternative to Francesco's nice geometric argument, the answer to question 1 is no, by definition of "hyperelliptic". I.e. a curve is hypereliptic if it has a line bundle of degree 2 with more than one section. If g = 3, any effective even theta charcteristic is such a line bundle. As to the subvariety M(1,g), this was (mostly) understood already by Riemann. I.e. if we look at the image of M(g) in the space of prin.pol. abelian varieties, then M(1,g) is just the intersection of Jacobians with the ample divisor defined by the vanishing of the theta function, hence it has pure codimension one, and is non empty. If g-1 is twice an odd number, the presence of hyperelliptics in this locus avoids the use of ampleness.