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Moduli of Cubic fourfolds and reducible OADP surfaces

Michele Bolognesi, Zakaria Brahimi, Hanine Awada

TL;DR

This work analyzes the moduli of cubic fourfolds by examining intersections of Hassett divisors $\mathcal{C}_8$ with $\mathcal{C}_{12}$ and $\mathcal{C}_{20}$. Using lattice-theoretic analysis, it enumerates the irreducible components of $\mathcal{C}_8\cap\mathcal{C}_{12}$ and $\mathcal{C}_8\cap\mathcal{C}_{20}$, characterized by discrete intersection data between a plane $P$ and a surface (scroll) $S$ or a Veronese surface $V$. It then studies rationality of generic cubics in these components via two approaches: constructing odd-degree multi-sections of the quadric fibration induced by projection off $P$, and producing OADP (one apparent double point) reducible surfaces inside $X$, with several components shown to be rational. Finally, the authors provide explicit cubic equations for representatives in each component using Macaulay2, together with explicit degenerations illustrating how the geometry of $S\cap P$ or $V\cap P$ governs component membership and rationality. The results deepen understanding of special cubic fourfolds and rationality phenomena in moduli, offering concrete constructions and computations to complement the lattice-theoretic framework.

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.

Moduli of Cubic fourfolds and reducible OADP surfaces

TL;DR

This work analyzes the moduli of cubic fourfolds by examining intersections of Hassett divisors with and . Using lattice-theoretic analysis, it enumerates the irreducible components of and , characterized by discrete intersection data between a plane and a surface (scroll) or a Veronese surface . It then studies rationality of generic cubics in these components via two approaches: constructing odd-degree multi-sections of the quadric fibration induced by projection off , and producing OADP (one apparent double point) reducible surfaces inside , with several components shown to be rational. Finally, the authors provide explicit cubic equations for representatives in each component using Macaulay2, together with explicit degenerations illustrating how the geometry of or governs component membership and rationality. The results deepen understanding of special cubic fourfolds and rationality phenomena in moduli, offering concrete constructions and computations to complement the lattice-theoretic framework.

Abstract

In this paper we explore the intersection of the Hassett divisor , parametrizing smooth cubic fourfolds containing a plane with other divisors . Notably we study the irreducible components of the intersections with and . 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 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 , or by finding examples of reducible one-apparent-double-point surfaces inside . Finally, via some Macaulay computations, we give explicit equations for cubics in each component.
Paper Structure (23 sections, 12 theorems, 23 equations)

This paper contains 23 sections, 12 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Description of the contents
  3. Acknowledgements:
  4. Preliminary results
  5. Hodge theory for cubic fourfolds
  6. The divisor $\mathcal{C}_8$ and the quadric surface fibration
  7. Generalized OADP varieties
  8. Intersection theory on a cubic fourfold
  9. Cubic hypersurfaces in $\mathcal{C}_{12}$
  10. Explicit examples of cubic fourfolds in the irreducible components of $\mathcal{C}_{12}\cap\mathcal{C}_8$
  11. $S\cap P$ is a finite set.
  12. $S\cap P$ is a conic.
  13. $S\cap P$ is a line of the ruling.
  14. $S\cap P$ is the directrix of $S$.
  15. Cubic hypersurfaces in $\mathcal{C}_{20}$
  16. ...and 8 more sections

Key Result

Theorem 1.1

(= Thm. cap812 and Lemma 12coincide ) There are three irreducible components of $\mathcal{C}_8\cap \mathcal{C}_{12}$ indexed by the value $P\cdot S =\epsilon \in\{1, 2, 3\}$, where $P$ is a plane and $S$ the class of a cubic rational normal scroll (that is $S\cdot S=7$ and $S\cdot h^2=3)$ contained

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Definition 2.2
  • Corollary 2.3
  • Proposition 2.4
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 13 more