[Paper Review] Hypersurfaces quartiques de dimension 3 : non rationalit\'e stable

Jean-Louis Colliot-Thélène, Alena Pirutka|arXiv (Cornell University)|Feb 17, 2014

Geometric Analysis and Curvature Flows95 citations

TL;DR

This paper establishes that very general smooth quartic threefolds over the complex numbers are not stably rational, by adapting and extending C. Voisin's specialization method to allow singular total spaces and relaxed conditions on exceptional divisors in resolutions. The key result is that the degree-zero Chow group of these threefolds is not universally isomorphic to Z, proving stable non-rationality and yielding examples defined over number fields via specialization arguments.

ABSTRACT

Inspir\'es par un argument de C. Voisin, nous montrons l'existence d'hypersurfaces quartiques lisses dans ${\bf P}^4_{\mathbb C}$ qui ne sont pas stablement rationnelles, plus pr\'ecis\'ement dont le groupe de Chow de degr\'e z\'ero n'est pas universellement \'egal \`a $\mathbb Z$. --- There are (many) smooth quartic hypersurfaces in ${\bf P}^4_{\mathbb C}$ which are not stably rational. More precisely, their degree zero Chow group is not universally equal to $\mathbb Z$. The proof uses a variation of a specialisation method due to C. Voisin.

Motivation & Objective

• To prove that very general smooth quartic threefolds in P^4 are not stably rational.
• To extend C. Voisin's specialization method to allow singular total spaces and weaker conditions on exceptional divisors in resolutions.
• To construct examples of non-stably rational quartic threefolds defined over number fields via specialization.
• To show that the degree-zero Chow group of such threefolds is not universally isomorphic to Z.
• To establish non-stable rationality over p-adic fields and number fields for certain cubic threefolds.

Proposed method

• Adapts Voisin's specialization argument by relaxing smoothness assumptions on the total space of a family of threefolds.
• Uses Fulton's specialization homomorphism for Chow groups of zero-cycles under mild hypotheses.
• Constructs a singular quartic threefold Y birational to an Artin-Mumford threefold, with a resolution Z →Y satisfying key cohomological conditions.
• Applies the specialization argument to show that if a general fiber were stably rational, then H^3(Z, Z) tors would vanish, contradicting Artin-Mumford's result.
• Employs the notion of universal CH0-triviality to link stable rationality to the structure of the Chow group of zero-cycles.
• Uses Galois cohomology and Brauer group computations (via cyclic algebras) to detect nontrivial torsion in H^3, particularly in the context of k-rational points and descent.

Experimental results

Research questions

• RQ1 Are very general smooth quartic threefolds in P^4 over C stably rational?
• RQ2 Can Voisin's specialization method be extended to singular total spaces and less restrictive exceptional divisor conditions?
• RQ3 Do there exist non-stably rational quartic threefolds defined over number fields?
• RQ4 Is the degree-zero Chow group of a very general quartic threefold universally isomorphic to Z?
• RQ5 Can the specialization method detect non-stable rationality for cubic threefolds over p-adic or number fields?

Key findings

• There exist smooth quartic threefolds over C that are not stably rational, as their degree-zero Chow group is not universally isomorphic to Z.
• The specialization argument yields examples of non-stably rational quartic threefolds defined over the algebraic closure of Q.
• The method applies to construct non-stably rational cubic threefolds over p-adic fields and number fields.
• The existence of a nontrivial Brauer group element in the function field of a singular model implies that the resolution has nontrivial H^3 torsion, obstructing stable rationality.
• The specialization argument shows that a general fiber of a family with an Artin-Mumford special fiber cannot be stably rational, due to the non-vanishing of torsion in H^3 of the resolution.
• The paper constructs a singular quartic threefold Y with a resolution Z →Y such that H^4(Z, Z) tors ≠ 0, satisfying the necessary conditions for the specialization argument to apply.