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On the unirationality of conic bundles with discriminant of degree eight

Alex Casarotti, Søren Gammelgaard, Alex Massarenti

Abstract

We study the unirationality of surface conic bundles $π\colon S\to\mathbb P^1$ over an arbitrary field $k$ with discriminant degree $d_S=8$, the first case beyond the del Pezzo range. We divide these surfaces in four families and produce explicit rational multisections via tangent constructions and Cremona transformations. Over $C_1$ fields we obtain Zariski dense loci of minimal, hence non $k$-rational, yet $k$-unirational conic bundles in each family; for one of the types we prove that the dense unirational locus is indeed Zariski open. Finally, we investigate the deformation theory of these conic bundles and how their unirationality behaves under specialization.

On the unirationality of conic bundles with discriminant of degree eight

Abstract

We study the unirationality of surface conic bundles over an arbitrary field with discriminant degree , the first case beyond the del Pezzo range. We divide these surfaces in four families and produce explicit rational multisections via tangent constructions and Cremona transformations. Over fields we obtain Zariski dense loci of minimal, hence non -rational, yet -unirational conic bundles in each family; for one of the types we prove that the dense unirational locus is indeed Zariski open. Finally, we investigate the deformation theory of these conic bundles and how their unirationality behaves under specialization.

Paper Structure

This paper contains 14 sections, 35 theorems, 230 equations.

Table of Contents

  1. Introduction
  2. Conic bundles, Enriques' criterion and Brauer groups
  3. Brauer groups and residues
  4. Conic bundles of type (2,1,1)
  5. Conic bundles of type (2,2,0)
  6. Conic bundles of type (3,1,0)
  7. Conic bundles of type (4,0,0)
  8. Deformations of the conic bundles and of the ambient projective bundles
  9. Infinitesimal deformations of conic bundles
  10. Degenerating $\mathcal{T}_{2,1,1}$ to $\mathcal{T}_{2,2,0}$
  11. Degenerating $\mathcal{T}_{2,2,0}$ to $\mathcal{T}_{3,1,0}$
  12. Degenerating $\mathcal{T}_{3,1,0}$ to $\mathcal{T}_{4,0,0}$
  13. Higher degree discriminants
  14. Computer algebra scripts

Key Result

Theorem 1.1

Let $k$ be an infinite perfect $C_1$–field of characteristic different from 2, and $\mathbb{P}_{(d_{0,0},d_{0,1},d_{0,2},d_{1,1},d_{1,2},d_{2,2})}$ be the parameter space for conic bundles of multidegree $(d_{0,0},d_{0,1},d_{0,2},d_{1,1},d_{1,2},d_{2,2})$. For each there exists a Zariski dense subset of $\mathbb{P}_{(d_{0,0},d_{0,1},d_{0,2},d_{1,1},d_{1,2},d_{2,2})}$ whose general point represent

Theorems & Definitions (84)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.6
  • proof
  • Definition 2.8
  • Remark 2.11
  • Remark 2.12: Lang’s theorem
  • ...and 74 more