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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
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