A simple criterion for the uniruledness of an orthogonal modular variety

TL;DR

The paper introduces a practical uniruledness criterion for orthogonal modular varieties, linking birational geometry to lattice invariants via a bound involving $b$, $D$, $N$, and the Fourier coefficient $c_{1,0}(E_{k,L})$ of a vector-valued Eisenstein series. By combining Kudla’s intersection formulas, Miyaoka–Mori bend-and-break, and canonical class analysis, it translates lattice data into negative Kodaira dimension in many cases. This yields broad uniruledness results, notably for almost all Nikulin–Vinberg moduli spaces of lattice-polarized K3 surfaces with $ ho\ge3$ (12 potential exceptions) and for the Kummer moduli at $d=13$, thereby advancing understanding of the birational geometry of orthogonal modular varieties. The findings provide a concrete, lattice-oriented toolkit for testing uniruledness across families of moduli spaces arising from K3 surfaces and related geometric objects.

Abstract

We exhibit a simple uniruledness criterion for general orthogonal modular varieties in terms of invariants of the corresponding lattice. As an application, we obtain the uniruledness of almost all Nikulin--Vinberg moduli spaces parameterizing projective K3 surfaces of Picard number at least 3 and fixed finite automorphism group.

Theorems & Definitions (19)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1: See Theorem 11 in BK01
• Remark 2.2
• Proposition 3.1: See BBFW24*Section 4
• proof
• Proposition 3.2: See BBFW24*Proposition 1.2
• proof
• Theorem 3.3: = Theorem \ref{['thm:int:main']}
• proof