Are there, for any $k$, smooth plane curves $C\subset\mathbb{P}^2$ of degree $d=2k$ over $\mathbb{C}$ such that the space of global sections of all odd theta characteristics on $C$ is one dimensional?
Some thoughts:
- In general the degree needs to be to be even, as for odd $d\geq5$ the line bundle $\mathcal{O}_C(\frac{d-3}{2})$ is an odd theta characteristic with larger dimensional space of global sections.
- It is true for $d=4$.
- In general, an odd theta characteristic gives rise to a plane curve $C'$ of degree $d-3$ such that the intersection divisor on $C$ is of the form $2D$ for some effective divisor $D$. The space of global sections of this theta characteristic can then be identified with the space of all homogeneous degree $d-3$ polynomials vanishing on $D$. Because $D$ has degree $\frac{1}{2}d(d-3)$ and the space of degree $d-3$ polynomials is $\frac{1}{2}d(d-3)+1$ it is not unreasonable to expect this space to be one-dimensional.
- For fixed genus, the statement is true for a general (not necessarily) plane curve of genus $g$.
