Uniruledness of orthogonal modular varieties
type: Item
instance of: scholarly article
title: Uniruledness of orthogonal modular varieties
zbMATH Open document ID: 1309.14030
DOI: 10.1090/S1056-3911-2014-00632-9
author: Klaus Hulek
author: V. Gritsenko
published in: Journal of Algebraic Geometry
publication date: 2014-10-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/1202.3361
Mathematics Subject Classification ID: 14J15
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 14M20
Mathematics Subject Classification ID: 11F23
Mathematics Subject Classification ID: 11F50
Mathematics Subject Classification ID: 11H56
zbMATH DE Number: 6353749
zbMATH Keywords: modular form
zbMATH Keywords: uniruledness
zbMATH Keywords: moduli space
zbMATH Keywords: lattice
MaRDI profile type: Publication
arXiv ID: 1202.3361
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OpenAlex ID: W2067487246
review text: Orthogonal modular varieties are quotients \({\mathcal F}_L(\Gamma)\) of the symmetric domain \({\mathcal D}(L)\) associated with an even integral lattice \(L\) of signature \((2,n)\) by an arithmetic subgroup \(\Gamma\) of the orthogonal group \({\text{O}}(L)\). They are quasi-projective varieties of dimension~\(n\) and arise in many geometric contexts, such as moduli spaces of polarised \(K3\) surfaces. These spaces are usually of general type, although that may be difficult to prove: cases to the contrary are generally associated with some in some way exceptional modular form for~\(\Gamma\).NEWLINENEWLINEIn this paper the authors show how to use suitable modular forms to show that, in a few very particular cases, \({\mathcal F}_L(\Gamma)\) is uniruled. The main geometric ingredient is the uniruledness criterion of Mori and Miyaoka, which says that a smooth projective variety \(V\) is unruled if there is an open subset \(U\) such that through any \(x\in U\) there passes a curve \(C\) with \(C.K_V<0\). The modular forms they use are reflective modular forms, i.e.\ modular forms \(F\) whose divisor on \({\mathcal D}(L)\) is supported on the fixed divisors of \(\pm\)reflections in \(\Gamma\). Such a modular form allows one to write the canonical class of a projective smooth model of \({\mathcal F}_L(\Gamma)\) in terms of the bundle of modular forms, the reflection divisors, the boundary of a compactification and the exceptional divisors of the chosen resolution of signularities. Using this calculation, the authors show that the Mori-Miyaoka criterion is applicable whenever the weight \(k\) of \(F\) satisfies \(k>mn\), where \(m\) is the greatest multiplicity (i.e.\ vanishing order) of \(F\) along a reflection divisor.NEWLINENEWLINEThe simplest case would be \(m=1\), but such forms (called strongly reflective) are very rare. The authors exploit some of the few known examples to show that certain moduli spaces of lattice-polarised \(K3\) surfaces are uniruled: these ar…
reviewed by: G. K. Sankaran
description: scientific article; zbMATH DE number 6353749