Uniruledness of orthogonal modular varieties

Abstract

A strongly reflective modular form with respect to an orthogonal group of signature ( 2 , n ) (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 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 K 3 \mathrm {K3} surfaces. Finally, we prove that the moduli space of Kummer surfaces associated to ( 1 , 21 ) (1,21) -polarised abelian surfaces is uniruled.