A theta-characteristic on a Riemann surface $\Sigma$ is a holomorphic line bundle $L$ such that $L \otimes L \cong \omega$, i.e. a square root of the cotangent bundle. There is a moduli space (stack) of such pairs $(\Sigma, L \in \mathrm{Pic}(\Sigma))$, which I'll denote $\widetilde{M}_g^{1/2}$ when $\Sigma$ has genus $g$, and I believe it is not unreasonable to call this the moduli space of theta-characteristics. This is related to the moduli space $M_g^{1/2}$ of spin Riemann surfaces in that there is a map $M_g^{1/2} \to \widetilde{M}_g^{1/2}$ and this is a $\mathbb{Z}/2$-gerbe.

Now, one can just as well define moduli spaces $\widetilde{M}_g^{1/r}$ and $M_g^{1/r}$ for any $r$, and the natural map is now a $\mathbb{Z}/r$-gerbe. The space $M_g^{1/r}$ classifies families of $r$-spin Riemann surfaces, and $\widetilde{M}_g^{1/r}$ classifies families of Riemann surfaces with a section of the fibrewise Picard which is an $r$-th root of the canonical section.

I am rather uninformed with respect to algebraic geometry, so my question is: What would an algebraic geometer call the moduli space $\widetilde{M}_g^{1/r}$ for $r > 2$?


I've seen many people referring to what you call $r$-spin Riemann surfaces as higher spin curves (sometimes with the $r$, sometimes suppressed), so my thought would be to call it the moduli stack of $r$-theta characteristics or of higher theta characteristics, if the $r$ isn't as important.

  • $\begingroup$ Ah, this is the convention I have been adopting. $\endgroup$ – Oscar Randal-Williams Dec 31 '10 at 11:06

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