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Moduli of theta-characteristics via Nikulin surfaces

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Abstract

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 a general Prym curve from R g is a section of a Nikulin surface if and only if g ≤ 7 and g ≠ 6. Furthermore, R 7 has the structure of a fibre space over the corresponding moduli space of polarized Nikulin surfaces. 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.

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Authors and Affiliations

  1. Institut Für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Gavril Farkas

  2. Dipartimento di Matematica, Universitá Roma Tre, Largo San Leonardo Murialdo 1, 00146, Rome, Italy

    Alessandro Verra

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  1. Gavril Farkas
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  2. Alessandro Verra
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Correspondence to Gavril Farkas.

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Farkas, G., Verra, A. Moduli of theta-characteristics via Nikulin surfaces. Math. Ann. 354, 465–496 (2012). https://doi.org/10.1007/s00208-011-0739-z

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  • DOI: https://doi.org/10.1007/s00208-011-0739-z

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