Uniruledness of orthogonal modular varieties

Gritsenko, V.; Hulek, K.

doi:10.1090/S1056-3911-2014-00632-9

Authors: V. Gritsenko and K. Hulek
Journal: J. Algebraic Geom. 23 (2014), 711-725
DOI: https://doi.org/10.1090/S1056-3911-2014-00632-9
Published electronically: February 21, 2014
MathSciNet review: 3263666
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Abstract | References | Additional Information

Abstract: A strongly reflective modular form with respect to an orthogonal group of signature $(2,n)$ determines a Lorentzian Kac–Moody algebra. We find a new geometric application of such modular forms: we prove that if the weight is larger than $n$, then the corresponding modular variety is uniruled. We also construct new reflective modular forms and thus provide new examples of uniruled moduli spaces of lattice polarised $\mathrm {K3}$ surfaces. Finally, we prove that the moduli space of Kummer surfaces associated to $(1,21)$-polarised abelian surfaces is uniruled.

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Additional Information

V. Gritsenko
Affiliation: Université Lille 1, Laboratoire Paul Painlevé, F-59655 Villeneuve d’Ascq, Cedex, France; and Institut Universitaire de France
MR Author ID: 219176
Email: Valery.Gritsenko@math.univ-lille1.fr

K. Hulek
Affiliation: Institut für Algebraische Geometrie, Leibniz Universität Hannover, D-30060 Hannover, Germany
MR Author ID: 89705
Email: hulek@math.uni-hannover.de

Received by editor(s): February 16, 2012
Received by editor(s) in revised form: August 24, 1012
Published electronically: February 21, 2014
Article copyright: © Copyright 2014 University Press, Inc.