[Paper Review] Some Loci of Rational Cubic Fourfolds

Michele Bolognesi, Francesco Russo+1 more|arXiv (Cornell University)|Apr 22, 2015

Algebraic Geometry and Number Theory12 references7 citations

TL;DR

This paper proves that all smooth cubic fourfolds in the divisor $ \mathcal{C}_{14}$ of the moduli space are rational by showing that every such fourfold contains a surface with one apparent double point (OADP surface), specifically a quartic rational normal scroll. The authors use a modern reformulation of Fano's classical results on rational maps defined by quadrics vanishing on such scrolls, and demonstrate that the existence of such a surface implies rationality, thereby establishing rationality for the entire divisor $ \mathcal{C}_{14}$, including non-Pfaffian examples.

ABSTRACT

In this paper we investigate the divisor $\\mathcal C_{14}$ inside the moduli space of smooth cubic hypersurfaces in $\\mathbb P^5$, whose generic element is a smooth cubic containing a smooth quartic scroll. Using the fact that all degenerations of quartic scrolls in $\\mathbb P^5$ contained in a smooth cubic hypersurface are surfaces with one apparent double point, we conclude that every cubic hypersurface belonging to $\\mathcal C_{14}$ is rational. As an application of our results and of the construction of some explicit examples contained in the Appendix, we also prove that the Pfaffian locus is not open in $\\mathcal C_{14}$.

Motivation & Objective

• To establish the rationality of all smooth cubic fourfolds in the divisor $ \mathcal{C}_{14}$, which parametrizes those containing a smooth quartic rational normal scroll.
• To provide a modern, detailed account of classical results from Fano (1943) on rational maps defined by quadrics vanishing on rational normal scrolls.
• To show that the existence of a surface with one apparent double point (OADP surface) in a cubic fourfold implies its rationality.
• To demonstrate that the Pfaffian locus is not open in $ \mathcal{C}_{14}$, resolving a key question about the structure of this divisor.
• To construct explicit examples of rational cubic fourfolds in $ \mathcal{C}_{14}$ containing both a smooth quartic rational normal scroll and two disjoint planes, using Macaulay2 computations.

Proposed method

• Utilizes the Hodge-theoretic characterization of $ \mathcal{C}_{14}$ to link integral cohomology classes $T \in \mathrm{H}^{2,2}(X,\mathbb{Z})$ with $T^2 = 10$ and $T \cdot h^2 = 4$ to degree-four surfaces with one apparent double point.
• Analyzes the rational map defined by the linear system of quadrics vanishing on a smooth quartic rational normal scroll embedded in $\mathbb{P}^5$, restricting it to cubic fourfolds containing the scroll.
• Applies Fano's classical results on the geometry of the rational map from a cubic fourfold to $\mathbb{P}^4$ induced by the linear system of quadrics vanishing on a scroll, reformulating them in modern algebraic geometry language.
• Studies the Hilbert scheme of quartic rational normal scrolls in cubic fourfolds in $ \mathcal{C}_{14}$, showing that all degenerations of such scrolls in a smooth cubic are surfaces with one apparent double point.
• Employs Macaulay2 to construct explicit examples of cubic fourfolds in $ \mathcal{C}_{14}$ containing a smooth quartic rational normal scroll and two disjoint planes, verifying the geometry via ideal computations.
• Uses birational geometry techniques, including the construction of a rational map from $\mathbb{P}^4$ to the cubic fourfold via the image of a quintic del Pezzo surface, to confirm rationality.

Experimental results

Research questions

• RQ1 Does every smooth cubic fourfold in the divisor $ \mathcal{C}_{14}$ admit a rational structure, and if so, what geometric condition guarantees this?
• RQ2 Can the rationality of the general member of $ \mathcal{C}_{14}$ be extended to all members via a geometric invariant, such as the presence of a surface with one apparent double point?
• RQ3 Is the Pfaffian locus open within $ \mathcal{C}_{14}$, or are there non-Pfaffian rational cubic fourfolds in this divisor?
• RQ4 What is the precise Hodge-theoretic characterization of surfaces of degree four with one apparent double point in a cubic fourfold?
• RQ5 Can explicit examples of cubic fourfolds in $ \mathcal{C}_{14}$ be constructed that contain both a smooth quartic rational normal scroll and two disjoint planes?

Key findings

• All smooth cubic fourfolds in the divisor $ \mathcal{C}_{14}$ are rational, extending the known rationality of general members to the entire divisor.
• Every class $T \in \mathrm{H}^{2,2}(X,\mathbb{Z})$ with $T^2 = 10$ and $T \cdot h^2 = 4$ on a smooth cubic fourfold is represented by a (possibly reducible) degree-four surface with one apparent double point.
• The Pfaffian locus is not open in $ \mathcal{C}_{14}$, as shown by the existence of non-Pfaffian rational cubic fourfolds in this divisor.
• Explicit examples of cubic fourfolds in $ \mathcal{C}_{14}$ are constructed that contain a smooth quartic rational normal scroll and two disjoint planes, with the residual intersection of a Segre threefold and the cubic hypersurface being a smooth rational normal scroll.
• The rational map from $\mathbb{P}^4$ to the cubic fourfold $V(F)$ is birational, as confirmed by Macaulay2 computations, providing a concrete birational parametrization of the variety.