Moduli of Cubic fourfolds and reducible OADP surfaces

TL;DR

This paper investigates the geometry and rationality properties of special cubic fourfolds containing a plane, focusing on their intersections with divisors parametrizing surfaces like cubic scrolls and Veronese surfaces, and provides explicit equations for these cases.

Contribution

It classifies the irreducible components of intersections of Hassett divisors in cubic fourfolds and analyzes their geometric and rationality properties, including explicit equations.

Findings

All irreducible components of the intersections with _{12} and _{20} are found.

The geometry of generic elements is described via intersections with surfaces.

Explicit equations for cubics in each component are provided.

Abstract

In this paper we explore the intersection of the Hassett divisor $\mathcal C_8$, parametrizing smooth cubic fourfolds $X$ containing a plane $P$ with other divisors $\mathcal C_i$. Notably we study the irreducible components of the intersections with $\mathcal{C}_{12}$ and $\mathcal{C}_{20}$. These two divisors generically parametrize respectively cubics containing a smooth cubic scroll, and a smooth Veronese surface. First, we find all the irreducible components of the two intersections, and describe the geometry of the generic elements in terms of the intersection of $P$ with the other surface. Then we consider the problem of rationality of cubics in these components, either by finding rational sections of the quadric fibration induced by projection off $P$, or by finding examples of reducible one-apparent-double-point surfaces inside $X$. Finally, via some Macaulay computations, we give explicit equations for cubics in each component.