VSPs of cubic fourfolds and the Gorenstein locus of the Hilbert scheme of 14 points on A^6

TL;DR

This paper investigates the geometry of the Gorenstein locus of the Hilbert scheme of 14 points on six-dimensional affine space, revealing its structure, smoothness, and relation to cubic fourfolds and secant varieties.

Contribution

It proves the smoothness of the exceptional component of the Gorenstein locus and identifies its intersection with the smoothable component as a vector bundle over a specific divisor in cubic fourfolds.

Findings

The exceptional component of the Gorenstein locus is smooth.

The intersection of components is a vector bundle over the Iliev-Ranestad divisor.

The ninth secant variety lies inside this divisor and is a codimension two complete intersection.

Abstract

This paper is concerned with the geometry of the Gorenstein locus of the Hilbert scheme of $14$ points on $\mathbb{A}^6$. This scheme has two components: the smoothable one and an exceptional one. We prove that the latter is smooth and identify the intersection of components as a vector bundle over the Iliev-Ranestad divisor in the space of cubic fourfolds. The ninth secant variety of the triple Veronese reembedding lies inside the Iliev-Ranestad divisor. We point out that this secant variety is set-theoretically a codimension two complete intersection and discuss the degrees of its equations.