[Paper Review] The Hasse principle for 9-nodal cubic threefolds

N. I. Shepherd‐Barron|arXiv (Cornell University)|Oct 18, 2012

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper establishes that for 9-nodal cubic threefolds over number fields, the only obstruction to the Hasse principle is the Brauer–Manin obstruction. Using descent theory, the authors construct universal torsors over the smooth locus, showing they are birational to cones over singular cubic 7-folds and satisfy the Hasse principle, thereby completing the classification of singular cubic threefolds with respect to the Hasse principle.

ABSTRACT

We prove that the Hasse principle holds for cubic threefolds with 9 singular points over a number field.

Motivation & Objective

• To complete the classification of singular cubic threefolds with respect to the Hasse principle by resolving the case of 9 nodes.
• To show that for 9-nodal cubic threefolds over number fields, the Brauer–Manin obstruction is the only obstruction to the existence of rational points.
• To extend the method of universal torsors and descent to singular cubic threefolds with high nodal count, using the Perazzo cubic fourfold as a geometric model.
• To demonstrate that universal torsors over the smooth locus of such threefolds are birational to cones over singular cubic 7-folds, enabling the application of the Hasse principle.
• To unify the treatment of singular cubic threefolds by relating them to Galois twists of the Perazzo cubic fourfold and exploiting its torus embedding structure.

Proposed method

• Use descent theory to construct universal torsors over the smooth locus $X^0$ of a 9-nodal cubic threefold $X$.
• Leverage the fact that $X$ is a hyperplane section of a Galois twist of the Perazzo cubic fourfold $P$, defined by $x_1x_2x_3 = y_1y_2y_3$, which is a torus embedding.
• Show that universal torsors over $X^0$ are $K$-birational to cones over singular cubic 7-folds, using the geometry of the Perazzo fourfold and its 9 three-planes.
• Apply results from [CT-Sal] on 3-nodal cubics and [CT-San2] on Brauer–Manin obstructions to deduce that the only obstruction to the Hasse principle is Brauer–Manin.
• Use the action of the Galois group on the 6 distinguished points of $P$ to define twists of $P$, and show that the corresponding torsors over $X^0$ are birational to line bundles over twisted Grassmannians.
• Establish that all such universal torsors satisfy the Hasse principle and weak approximation, relying on the rationality and homogeneity of the twisted Grassmannian $Gr(2,V_\rho)$.

Experimental results

Research questions

• RQ1 Does the Hasse principle hold for 9-nodal cubic threefolds over number fields, and if not, what is the obstruction?
• RQ2 Can the method of universal torsors be extended to singular cubic threefolds with 9 nodes, given their geometric complexity?
• RQ3 Is the Brauer–Manin obstruction the only obstruction to rational points on 9-nodal cubic threefolds, as it is for 3- and 6-nodal cases?
• RQ4 How does the geometry of the Perazzo cubic fourfold facilitate the construction of universal torsors over its hyperplane sections?
• RQ5 What is the role of Galois twists of the Perazzo fourfold in parametrizing 9-nodal cubic threefolds and their torsors?

Key findings

• For 9-nodal cubic threefolds over number fields, the Hasse principle holds if and only if the Brauer–Manin obstruction vanishes.
• Universal torsors over the smooth locus of a 9-nodal cubic threefold are $K$-birational to cones over singular cubic 7-folds.
• All such universal torsors satisfy the Hasse principle and weak approximation, due to their birational equivalence to line bundles over twisted Grassmannians.
• The smooth locus of a 9-nodal cubic threefold is a torsor under a 4-dimensional torus, and its class group is torsion-free of rank 5.
• The construction relies on the Perazzo cubic fourfold as a model, whose 9 three-planes and 9 singular lines form a $ ext{Gal}(K)$-invariant configuration under the wreath product $S_3 times S_2$.
• The 9-nodal cubic threefold arises as a hyperplane section of a Galois twist of the Perazzo fourfold, and its rational points are controlled by the Galois action on the 6 distinguished points of the fourfold.