On conjugacy classes of the Klein simple group in Cremona group

Hamid Abban

On conjugacy classes of the Klein simple group in Cremona group

Algebraic Geometry 2022-07-22 v2

Authors: Hamid Abban

Abstract

We consider countably many three dimensional $\mathtt{PSL}_2(\mathbb{F}_7)$ -del Pezzo surface fibrations over $\mathbb{P}^1$ . Conjecturally they are all irrational except two families, one of which is the product of a del Pezzo surface with $\mathbb{P}^1$ . We show that the other model is $\mathtt{PSL}_2(\mathbb{F}_7)$ -equivariantly birational to $\mathbb{P}^2\times\mathbb{P}^1$ . Based on a result of Prokhorov, we show that they are non-conjugate as subgroups of the Cremona group $\mathtt{Cr}_3(\mathbb{C})$ .

Keywords

conjugacy classes Klein simple group Cremona group del Pezzo surface fibrations rationality equivariant birational geometry PSL2(F7)

Cite

@article{arxiv.1310.5548,
  title  = {On conjugacy classes of the Klein simple group in Cremona group},
  author = {Hamid Abban},
  journal= {arXiv preprint arXiv:1310.5548},
  year   = {2022}
}

Comments

The results and content of the older version has been sharpened. Some parts have been published separately, and this is an update for section 3 in V1. Final version, to appear in Glasgow Math Journal

On conjugacy classes of the Klein simple group in Cremona group

Abstract

Keywords

Cite

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