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Automorphism groups of Mori Del Pezzo fibrations over an irrational curve

Pascal Fong, Susanna Zimmermann

Abstract

We study the automorphism groups of Mori Del Pezzo fibrations over a smooth projective curve $C$ of positive genus. From that, we obtain a classification of maximal connected algebraic subgroups of $\mathrm{Bir}(C\times \mathbb{P}^2)$. Our results hold over any algebraically closed field of characteristic zero.

Automorphism groups of Mori Del Pezzo fibrations over an irrational curve

Abstract

We study the automorphism groups of Mori Del Pezzo fibrations over a smooth projective curve of positive genus. From that, we obtain a classification of maximal connected algebraic subgroups of . Our results hold over any algebraically closed field of characteristic zero.

Paper Structure

This paper contains 35 sections, 92 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Plan of the article
  4. Notations and conventions
  5. Preliminaries
  6. Rationally connectedness and sections
  7. Repetition of Del Pezzo surfaces
  8. Generalities on Mori Del Pezzo fibrations
  9. Sarkisov links starting from Mori Del Pezzo fibrations
  10. Mori Del Pezzo fibrations of degree $8$
  11. When the vertical automorphism group is finite
  12. Description of the automorphism group
  13. The case of Mori Del Pezzo fibrations of degree $1,2,3,5$
  14. Mori Del Pezzo fibrations of degree $4$
  15. Mori Del Pezzo fibrations of degree $6$
  16. ...and 20 more sections

Key Result

Theorem A

Let $\pi\colon X\longrightarrow C$ be a Mori Del Pezzo fibration above a smooth projective curve $C$ of genus $g(C)\geq1$ and let $d=K_{X_{\mathbf{k}(C)}}^2$. If $\mathrm{Aut}^{\circ}(X)$ is a maximal connected algebraic subgroup of $\mathop{\mathrm{Bir}}\nolimits(X)$, then exactly one of the follow

Theorems & Definitions (196)

  • Theorem A
  • Theorem B
  • Theorem C
  • Proposition 1: \ref{['pro:DP5']}
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Remark 1
  • Lemma 1: GHS02
  • Lemma 2: GHS02
  • ...and 186 more