Uniruledness of orthogonal modular varieties (Q2922503)

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 are the cases \(L=2U\oplus S(-1)\) for \(U\) the hyperbolic plane and \(S\) in a short list of positive definite lattices, and \(\Gamma\) a suitable subgroup. For some other cases there are further results with a different subgroup. Moreover, they exhibit a case with \(m>1\) where their results can be applied, showing that the moduli space of Kummer surfaces associated with \((1,21)\)-polarised abelian surfaces is uniruled.