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Automorphism groups of $\mathbb{P}^1$-bundles over ruled surfaces

Pascal Fong

Abstract

We classify the pairs $(X,π)$, where $π\colon X\to S$ is a $\mathbb{P}^1$-bundle over a non-rational ruled surface $S$ and $\mathrm{Aut}^\circ(X)$ is relatively maximal, i.e., maximal with respect to the inclusion in the group $\mathrm{Bir}(X/S)$. The results hold over any algebraically closed field of characteristic zero.

Automorphism groups of $\mathbb{P}^1$-bundles over ruled surfaces

Abstract

We classify the pairs , where is a -bundle over a non-rational ruled surface and is relatively maximal, i.e., maximal with respect to the inclusion in the group . The results hold over any algebraically closed field of characteristic zero.
Paper Structure (36 sections, 75 theorems, 183 equations)

This paper contains 36 sections, 75 theorems, 183 equations.

Table of Contents

  1. Introduction
  2. Automorphism groups of ruled surfaces
  3. Reduction via the Minimal Model Program
  4. The main results
  5. Plan of the article
  6. Preliminaries
  7. Relatively maximal automorphism groups, (super)stiffness, and Blanchard's Lemma
  8. Geometrically ruled surfaces and maximal connected algebraic subgroups of $\mathrm{Bir}(C\times \mathbb{P}^1)$
  9. First reduction step: towards $\mathbb{F}_b$-bundles
  10. Removal of jumping fibers
  11. Fiber products of ruled surfaces as $\mathbb{F}_0$-bundles
  12. Transition maps of $\mathbb{F}_b$-bundles for $b>0$
  13. Sarkisov Program for $\mathbb{F}_b$-bundles
  14. Second reduction step: towards $\mathbb{F}_b$-bundles without invariant fibers
  15. $\mathbb{F}_b$-bundles with invariant fibers
  16. ...and 21 more sections

Key Result

Theorem A

Let $\mathbf{k}$ be an algebraically closed field of characteristic zero. Let $C$ be a smooth projective curve of genus $g\geq 1$, and let $\tau \colon S\to C$ and $\pi\colon X\to S$ be $\mathbb{P}^1$-bundles.

Theorems & Definitions (161)

  • Theorem A
  • Proposition B
  • Theorem C
  • Corollary D
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • Corollary 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 151 more