Tri-Coble surfaces and their automorphisms

John Lesieutre

doi:10.3934/jmd.2021008

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2021, Volume 17: 267-284. Doi: 10.3934/jmd.2021008

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Tri-Coble surfaces and their automorphisms

John Lesieutre

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA

Received: April 2020

Revised: January 28, 2021

Published: May 2021

This work was supported by NSF grant DMS-1912476 and a Sloan Research Fellowship

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Abstract

We construct some positive entropy automorphisms of rational surfaces with no periodic curves. The surfaces in question, which we term tri-Coble surfaces, are blow-ups of $ \mathbb P^2$ at 12 points which have contractions down to three different Coble surfaces. The automorphisms arise as compositions of lifts of Bertini involutions from certain degree $1$ weak del Pezzo surfaces.

Keywords:

Automorphisms of algebraic surfaces,
Coble surfaces.

Mathematics Subject Classification: Primary: 14J50; Secondary: 37F10.

Citation:

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Figure 1. A configuration of three tangent quadrics. In this case, two of the quadrics are tangent along an entire curve

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Figure 2. $ \tau_ \mathbf p $ fixes $ \mathbf q $ and $ \tau_ \mathbf q $ fixes $ \mathbf p $ if and only if $ S $ is tangent to four conic curves. According to Lemma 2.6, this is possible only if $ S $ is tangent to a conic surface, which means that $ S_{ \mathbf p \mathbf q} $ is a Coble surface

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Figure 3. The action of $ G $ on $ \Delta $

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Figure 4. The set $ {\rm{Conv}}(\Lambda) $

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