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The geometry of the moduli space of odd spin curves. (English) Zbl 1325.14045
A theta characteristic on a smooth curve $$C$$ of genus $$g$$ is a line bundle $$\eta \in \mathrm{Pic}(C)$$ such that $$\eta^{\otimes 2} \cong \omega_C$$. The theta characteristic is called odd resp. even according to the parity of $$h^0(C,\, \eta)$$. The stack $$\mathbf{S}_g$$ of smooth curves with a theta characteristic is a smooth Deligne-Mumford stack admitting a natural forgetful functor to $$\mathbf{M}_g$$. One can construct a compactification $$\overline{\mathbf{S}}_g \to \overline{\mathbf{M}}_g$$ by letting smooth curves with a theta characteristic degenerate to so-called spin curves. This stack consists of two connected components $$\overline{\mathbf{S}}_g^+$$ and $$\overline{\mathbf{S}}_g^-$$ corresponding to even and odd theta characteristics, respectively. The even case was treated in [G. Farkas, Adv. Math. 223, No. 2, 433–443 (2010; Zbl 1183.14020)] and [G. Farkas and A. Verra, Math. Ann. 354, No. 2, 465–496 (2012; Zbl 1259.14033)], while the subject of the present paper is the birational geometry of $$\overline{\mathcal{S}}_g^-$$, the coarse moduli space associated to $$\overline{\mathbf{S}}_g^-$$.
The obtained results are quite comprehensive: The authors show that $$\overline{\mathcal{S}}_g^-$$ is a variety of general type for $$g \geq 12$$ and uniruled (hence of negative Kodaira dimension) for $$g \leq 11$$. Moreover they show that for $$g \leq 8$$ these spaces are even unirational, and provide explicit birational models.
The covering rational curves needed for the uniruledness result are obtained as theta pencils on $$K3$$ surfaces, which essentially are rational families of hyperplane sections of a $$K3$$ surface that contain the support of a fixed odd theta characteristic of any of its member curves. This construction can be carried out for $$g \leq 9$$ and $$g = 11$$, as the general curves of these genera lie on a $$K3$$ surface. For $$g = 10$$ the approach has to be modified somewhat using the normalization map for irreducible $$1$$-nodal curves of arithmetic genus $$11$$.
Unirationality stems from the fact that the general curve of genus $$g \leq 6$$ has a plane sextic model, while for $$g = 7$$ and $$8$$ the Mukai models are used. The latter are obtained from certain homogeneous spaces on which the general curve of genus $$7$$ up to $$10$$ can be obtained as a linear section. These models are used to construct a unirational variety that dominates $$\overline{\mathcal{S}}_g^-$$.
The fact that $$\overline{\mathcal{S}}_g^-$$ is of general type for $$g \geq 12$$ is obtained by constructing the divisor of degenerate theta characteristics and computing its class. Together with the classes of well-known divisors of small slope this can then be used to deduce that $$K_{\overline{\mathcal{S}}_g^-}$$ is big. For the latter one can use a Brill-Noether divisor if $$g \geq 13$$, while for $$g = 12$$ a special divisor has to be constructed that constitutes a counterexample to the slope conjecture of J. Harris and I. Morrison [Invent. Math. 99, No. 2, 321–355 (1990; Zbl 0705.14026)].

##### MSC:
 14H10 Families, moduli of curves (algebraic) 14D23 Stacks and moduli problems 14E08 Rationality questions in algebraic geometry 14M20 Rational and unirational varieties
##### Software:
KoszulDivisorOnPic14M8
Full Text:
##### References:
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