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On the rationality of some real threefolds

Olivier Benoist, Alena Pirutka

Abstract

We study the rationality of some geometrically rational three-dimensional conic and quadric surface bundles, defined over the reals and more general real closed fields, for which the real locus is connected and the intermediate Jacobian obstructions to rationality vanish. We obtain both negative and positive results, using unramified cohomology and birational rigidity techniques, as well as concrete rationality constructions.

On the rationality of some real threefolds

Abstract

We study the rationality of some geometrically rational three-dimensional conic and quadric surface bundles, defined over the reals and more general real closed fields, for which the real locus is connected and the intermediate Jacobian obstructions to rationality vanish. We obtain both negative and positive results, using unramified cohomology and birational rigidity techniques, as well as concrete rationality constructions.

Paper Structure

This paper contains 13 sections, 24 theorems, 21 equations.

Table of Contents

  1. Degree $3$ unramified cohomology
  2. A nonrational threefold over a nonarchimedean real closed field
  3. Consequences over the reals
  4. Rationality of some degree $4$ conic bundles
  5. Preliminaries for the Sarkisov program over nonclosed fields
  6. Running the MMP over nonclosed fields
  7. Mori fiber spaces
  8. Sarkisov links
  9. Conic bundles
  10. Birational rigidity over nonclosed fields
  11. Running the Sarkisov program
  12. Birational rigidity of conic bundles
  13. An application

Key Result

Theorem 1

For each even $d\geq 2$, there exists a variety $X$ with equation (eq1) over the real closed field $R:=\cup_{n\geq 1}\mathbb R((t^{\frac{1}{n}}))$ that is not $R$-rational.

Theorems & Definitions (49)

  • Theorem 1: Theorem \ref{['th1']}
  • Corollary 2: Corollary \ref{['correal']}
  • Theorem 3: Theorem \ref{['th2']}
  • Theorem 4: Theorem \ref{['th3']}
  • Theorem 5: Theorem \ref{['th4']}
  • Theorem 1.1
  • Remark 1.2
  • Proposition 1.3
  • proof
  • proof : Proof of Theorem \ref{['th1']}
  • ...and 39 more