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Moduli of theta-characteristics via Nikulin surfaces

We study moduli spaces of K3 surfaces endowed with a Nikulin involution and their image in the moduli space R_g of Prym curves of genus g. We observe a striking analogy with Mukai's well-known work on ordinary K3 surfaces. Many of Mukai's results have a very precise Prym-Nikulin analogue, for instance: (1) A general Prym curve from R_g is a section of a Nikulin surface if and only if g\leq 7, g\neq 6. (2) R_7 has the structure of a Mori fibre space over the corresponding moduli space of polarized Nikulin surfaces. (3) In the case of genus next to maximal (g=6), the Prym-Nikulin locus in R_6 is an extremal divisor; the corresponding space of Nikulin surfaces has a Grassmannian GIT model. (4) The Prym-Nikulin locus in R_g can be characterized by extra syzygies of Prym-canonical curves. We then use these results to study the geometry of the moduli space S_g of even spin curves, with special emphasis on the transition case of S_8 which is a 21-dimensional Calabi-Yau variety.