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Torsion order and irrationality of complete intersections

Jan Lange, Guoyun Zhang

TL;DR

The paper studies the torsion order Tor(X) of zero-cycles as an obstruction to rationality and develops a framework of affine degenerations to transfer torsion bounds from hypersurfaces to complete intersections. Using Gröbner-basis controlled degenerations and strictly semi-stable two-component fibres, the authors derive new logarithmic lower bounds for Tor(X) for very general complete intersections in projective spaces, as well as for hypersurfaces in products of projective spaces and Grassmannians. Consequences include divisibility by a prescribed m (and in some cases by 2) of Tor(X), implying non-rationality, non-stable rationality, and lack of A1-connectedness in broad families. The results generalize and extend prior motivic and unramified-cohomology obstructions, broadening the scope of varieties for which irrationality can be detected via torsion in CH0 and related diagonal decompositions.

Abstract

We provide new logarithmic lower bounds for the torsion order of a very general complete intersection in projective space as well as a very general hypersurface in products of projective spaces and Grassmannians, in particular we prove their retract irrationality.

Torsion order and irrationality of complete intersections

TL;DR

The paper studies the torsion order Tor(X) of zero-cycles as an obstruction to rationality and develops a framework of affine degenerations to transfer torsion bounds from hypersurfaces to complete intersections. Using Gröbner-basis controlled degenerations and strictly semi-stable two-component fibres, the authors derive new logarithmic lower bounds for Tor(X) for very general complete intersections in projective spaces, as well as for hypersurfaces in products of projective spaces and Grassmannians. Consequences include divisibility by a prescribed m (and in some cases by 2) of Tor(X), implying non-rationality, non-stable rationality, and lack of A1-connectedness in broad families. The results generalize and extend prior motivic and unramified-cohomology obstructions, broadening the scope of varieties for which irrationality can be detected via torsion in CH0 and related diagonal decompositions.

Abstract

We provide new logarithmic lower bounds for the torsion order of a very general complete intersection in projective space as well as a very general hypersurface in products of projective spaces and Grassmannians, in particular we prove their retract irrationality.

Paper Structure

This paper contains 23 sections, 29 theorems, 192 equations.

Table of Contents

  1. Introduction
  2. Complete intersections
  3. Hypersurfaces in rational varieties
  4. Outline of the argument
  5. Preliminaries
  6. Notations and conventions
  7. Monomial orderings and Gröbner basis
  8. Strictly semi-stable schemes
  9. Relative torsion orders
  10. Very general elements of a family
  11. Affine degenerations
  12. Strongly rational algebras
  13. Affine strictly semi-stable families
  14. The key construction
  15. Admissible pairs and torsion orders
  16. ...and 8 more sections

Key Result

Theorem 1.1

The torsion order of a very general complete intersection of multidegree $(d_1,\dots,d_s)$ and positive Fano index $r$ over a field $k$ is divisible by a positive integer $m$ if $m \in k^\ast$ and there exists a degree $d_i$ such that $d_i \geq \log_{2} (r + m) + m$.

Theorems & Definitions (80)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Example 2.1: CLO15
  • Example 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Definition 2.5
  • ...and 70 more