Variety of power sums and divisors in the moduli space of cubic fourfolds

TL;DR

This paper investigates the geometric properties of certain cubic fourfolds, showing their associated varieties of power sums are singular along a K3 surface and identifying a special divisor in the moduli space.

Contribution

It demonstrates that cubic fourfolds apolar to a Veronese surface form a unique divisor in the moduli space, distinct from Noether-Lefschetz divisors, and explores their Hodge-theoretic implications.

Findings

VSP(F,10) is singular along a K3 surface of genus 20.

These cubic fourfolds form a divisor in the moduli space.

No nontrivial Hodge correspondence exists between a very general cubic and its VSP.

Abstract

We show that a cubic fourfold F that is apolar to a Veronese surface has the property that its variety of power sums VSP(F,10) is singular along a K3 surface of genus 20. We prove that these cubics form a divisor in the moduli space of cubic fourfolds and that this divisor is not a Noether-Lefschetz divisor. We use this result to prove that there is no nontrivial Hodge correspondence between a very general cubic and its VSP.