Finite generation of Noether-Lefschetz divisors and the slope of the moduli space of cubic fourfolds

TL;DR

The article develops a quantitative, modular-forms based understanding of divisor geometry on moduli spaces of cubic fourfolds and lattice-polarized K3 surfaces. It proves a finite, uniform presentation of the rational Picard group for ${\\mathcal{F}}_{2d}$ by showing it is generated by Noether–Lefschetz divisors of discriminant at most $4d$ and gives explicit expressions for the Hodge class in terms of NL divisors via vector-valued Eisenstein series and theta-series coefficients. It also introduces a slope invariant for the cubic fourfold moduli, establishing sharp bounds $523{,}777/206{,}215{,}591 \le s(\\mathcal{M}) \le 16/9$, and demonstrates a parallel slope result for K3 degree $2$ moduli. The framework relies on the Kudla–Millson theta correspondence for orthogonal Shimura varieties, providing a unified method to study effective divisors and higher-codimension NL cycles with broad applicability to lattice-polarized hyperkähler varieties and related moduli spaces.

Abstract

We study divisors on moduli spaces of cubic fourfolds with simple singularities and of quasi-polarized K3 surfaces of degree $2d$. For the moduli space of cubic fourfolds, we introduce a slope quantity to characterize the effective cone and prove an explicit bound for it. For the K3 moduli spaces, we give an explicit finite presentation of the rational Picard group by showing that it is generated by Noether-Lefschetz divisors of discriminant less than or equal to $4d$. As a byproduct, we obtain two explicit expressions for the Hodge class in terms of Noether-Lefschetz divisors, and we indicate analogous results for higher-codimension Noether-Lefschetz cycles.

Theorems & Definitions (29)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• Proposition 3.1
• proof
• Remark 3.2
• Theorem 3.3: Bruinier--Westerholt-Raum
• proof