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Automorphism groups and linearizability of rational Fano conic bundle threefolds

Shuto Abe

TL;DR

This work extends the equivariant intermediate Jacobian torsor obstruction to algebraically closed fields of characteristic zero and uses it to analyze projective linearizability of finite group actions on rational threefolds. It develops the theory of $G$-equivariant principally polarized abelian varieties and Chow group schemes, and constructs the equivariant intermediate Jacobian torsor $\mathrm{IJ}(X)$ to compare $G$-invariant pieces of Chow groups with Picard data of curves. A central contribution is the verification that the obstruction vanishes for projectively linearizable actions, and the use of equivariant Prym schemes to connect Chow groups with Prym varieties, enabling linearizability results for conic-bundle Fano threefolds in Mori–Mukai № 2.18. The automorphism groups of smooth Fano threefolds № 2.18 are analyzed, with a general member shown to have $\mathrm{Aut}(X) \cong \mathbb{Z}/2\mathbb{Z}$ and a bound $|\mathrm{Aut}(X)| \le 336$; as an application, a general such threefold is proved to be $\mathrm{Aut}(X)$-linearizable, while explicit non-linearizable actions are also demonstrated.

Abstract

We generalize the equivariant intermediate Jacobian torsor obstruction over $\mathbb{C}$ to algebraically closed fields of characteristic zero. It is an obstruction to the (projective) linearizability problem of finite group actions on threefolds. In addition, we calculate automorphism groups of general smooth Fano threefolds of No. 2.18. As an application, we prove that a general smooth Fano threefold X of No. 2.18 is linearizable for its automorphism group.

Automorphism groups and linearizability of rational Fano conic bundle threefolds

TL;DR

This work extends the equivariant intermediate Jacobian torsor obstruction to algebraically closed fields of characteristic zero and uses it to analyze projective linearizability of finite group actions on rational threefolds. It develops the theory of -equivariant principally polarized abelian varieties and Chow group schemes, and constructs the equivariant intermediate Jacobian torsor to compare -invariant pieces of Chow groups with Picard data of curves. A central contribution is the verification that the obstruction vanishes for projectively linearizable actions, and the use of equivariant Prym schemes to connect Chow groups with Prym varieties, enabling linearizability results for conic-bundle Fano threefolds in Mori–Mukai № 2.18. The automorphism groups of smooth Fano threefolds № 2.18 are analyzed, with a general member shown to have and a bound ; as an application, a general such threefold is proved to be -linearizable, while explicit non-linearizable actions are also demonstrated.

Abstract

We generalize the equivariant intermediate Jacobian torsor obstruction over to algebraically closed fields of characteristic zero. It is an obstruction to the (projective) linearizability problem of finite group actions on threefolds. In addition, we calculate automorphism groups of general smooth Fano threefolds of No. 2.18. As an application, we prove that a general smooth Fano threefold X of No. 2.18 is linearizable for its automorphism group.

Paper Structure

This paper contains 14 sections, 44 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Equivariant principally polarized abelian varieties over algebraically closed fields
  3. Equivariant intermediate Jacobian torsor obstructions
  4. Murre's intermediate Jacobians
  5. Equivariant Murre's intermediate Jacobians
  6. Chow group schemes
  7. Obstructions of linearizability
  8. Automorphism groups of smooth Fano threefolds of № 2.18
  9. Fano threefolds of № 2.18
  10. Family of smooth Fano threefolds of № 2.18
  11. Linearizability
  12. Equivariant polarized Prym schemes and Prym varieties
  13. Intermediate Jacobians and Prym varieties
  14. Linearizability of Fano threefolds of № 2.18

Key Result

Theorem 1.2

(Theorem IJT) Let $X$ be a smooth projective rational threefold over an algebraically closed field $k$ of characteristic zero with a regular and projectively linearizable action of a finite group $G$. Then there exists a smooth projective curve $C$ with a regular $G$-action such that, for any $G$-in

Theorems & Definitions (92)

  • Definition 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Definition 2.1
  • Example 2.2
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • ...and 82 more