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Geometric Invariant Theory of Syzygies, with Applications to Moduli Spaces

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Geometry of Moduli (Abelsymposium 2017)

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Abstract

We define syzygy points of projective schemes, and introduce a program of studying their GIT stability. Then we describe two cases where we have managed to make some progress in this program, that of polarized K3 surfaces of odd genus, and of genus six canonical curves. Applications of our results include effectivity statements for divisor classes on the moduli space of odd genus K3 surfaces, and a new construction in the Hassett-Keel program for the moduli space of genus six curves.

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Acknowledgements

Foremost, this paper owes its existence to the organizers of the Abel Symposium 2017 “Geometry of Moduli,” who gave me an opportunity and motivation to write up this work. I am also indebted to Gavril Farkas, whose influence is evident in every section of this paper, and who generously shared his and Seán Keel’s ideas to use syzygies as the means to construct the canonical model of \( \operatorname {\overline {\mathrm {M}}}_g\) at an AIM workshop in December 2012. All results in this paper grew out of my attempt to implement these ideas. I am grateful to Anand Deopurkar for his comments and suggestions on an earlier version of this paper. During the preparation of this paper, I was partially supported by the NSA Young Investigator grant H98230-16-1-0061 and Alfred P. Sloan Research Fellowship.

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Authors and Affiliations

  1. Department of Mathematics, Boston College, Chestnut Hill, MA, USA

    Maksym Fedorchuk

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  1. Maksym Fedorchuk
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Correspondence to Maksym Fedorchuk .

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Editors and Affiliations

  1. Department of Mathematics, University of Oslo, Oslo, Norway

    Jan Arthur Christophersen

  2. Department of Mathematics, University of Oslo, Oslo, Norway

    Kristian Ranestad

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Fedorchuk, M. (2018). Geometric Invariant Theory of Syzygies, with Applications to Moduli Spaces. In: Christophersen, J., Ranestad, K. (eds) Geometry of Moduli. Abelsymposium 2017. Abel Symposia, vol 14. Springer, Cham. https://doi.org/10.1007/978-3-319-94881-2_5

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