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

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

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

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: John Lesieutre. Tri-Coble surfaces and their automorphisms. Journal of Modern Dynamics, 2021, 17: 267-284. doi: 10.3934/jmd.2021008
References:
[1] H. F. Baker, Principles of Geometry. Volume 6. Introduction to the Theory of Algebraic Surfaces and Higher Loci, Cambridge Library Collection, Cambridge University Press, Cambridge, 2010.   Google Scholar
[2]

E. Bedford and K. Kim, Periodicities in linear fractional recurrences: Degree growth of birational surface maps, Michigan Math. J., 54 (2006), 647-670.  doi: 10.1307/mmj/1163789919.  Google Scholar

[3]

E. Bedford and K. Kim, Dynamics of rational surface automorphisms: Linear fractional recurrences, J. Geom. Anal., 19 (2009), 553-583.  doi: 10.1007/s12220-009-9077-8.  Google Scholar

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E. Bedford and K. Kim, Continuous families of rational surface automorphisms with positive entropy, Math. Ann., 348 (2010), 667-688.  doi: 10.1007/s00208-010-0498-2.  Google Scholar

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J. Blanc, Dynamical degrees of (pseudo)-automorphisms fixing cubic hypersurfaces, Indiana Univ. Math. J., 62 (2013), 1143-1164.  doi: 10.1512/iumj.2013.62.5040.  Google Scholar

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A. B. Coble, The {t}en {n}odes of the {r}ational {s}extic and of the {C}ayley {s}ymmetroid, Amer. J. Math., 41 (1919), 243-265.  doi: 10.2307/2370285.  Google Scholar

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J. Lesieutre, Code for "{T}ri-{C}oble surfaces and their automorphisms'', ScholarSphere, Penn State University Libraries, 2021. doi: 10.26207/xwyq-hc68.  Google Scholar

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B. Maskit, On {P}oincaré's theorem for fundamental polygons, Advances in Math., 7 (1971), 219-230.  doi: 10.1016/S0001-8708(71)80003-8.  Google Scholar

[10]

C. T. McMullen, Dynamics on blowups of the projective plane, Publ. Math. Inst. Hautes Études Sci., 105 (2007), 49–89. doi: 10.1007/s10240-007-0004-x.  Google Scholar

[11]

The Sage Developers, SageMath, the Sage Mathematics Software System (Version 9.2), 2020., Available at https://www.sagemath.org. Google Scholar

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T. Uehara, Rational surface automorphisms with positive entropy, Ann. Inst. Fourier (Grenoble), 66 (2016), 377-432.  doi: 10.5802/aif.3014.  Google Scholar

show all references

References:
[1] H. F. Baker, Principles of Geometry. Volume 6. Introduction to the Theory of Algebraic Surfaces and Higher Loci, Cambridge Library Collection, Cambridge University Press, Cambridge, 2010.   Google Scholar
[2]

E. Bedford and K. Kim, Periodicities in linear fractional recurrences: Degree growth of birational surface maps, Michigan Math. J., 54 (2006), 647-670.  doi: 10.1307/mmj/1163789919.  Google Scholar

[3]

E. Bedford and K. Kim, Dynamics of rational surface automorphisms: Linear fractional recurrences, J. Geom. Anal., 19 (2009), 553-583.  doi: 10.1007/s12220-009-9077-8.  Google Scholar

[4]

E. Bedford and K. Kim, Continuous families of rational surface automorphisms with positive entropy, Math. Ann., 348 (2010), 667-688.  doi: 10.1007/s00208-010-0498-2.  Google Scholar

[5]

J. Blanc, Dynamical degrees of (pseudo)-automorphisms fixing cubic hypersurfaces, Indiana Univ. Math. J., 62 (2013), 1143-1164.  doi: 10.1512/iumj.2013.62.5040.  Google Scholar

[6]

A. B. Coble, The {t}en {n}odes of the {r}ational {s}extic and of the {C}ayley {s}ymmetroid, Amer. J. Math., 41 (1919), 243-265.  doi: 10.2307/2370285.  Google Scholar

[7] I. V. Dolgachev, Classical Algebraic Geometry: A Modern View, Cambridge University Press, Cambridge, 2012.  doi: 10.1017/CBO9781139084437.  Google Scholar
[8]

J. Lesieutre, Code for "{T}ri-{C}oble surfaces and their automorphisms'', ScholarSphere, Penn State University Libraries, 2021. doi: 10.26207/xwyq-hc68.  Google Scholar

[9]

B. Maskit, On {P}oincaré's theorem for fundamental polygons, Advances in Math., 7 (1971), 219-230.  doi: 10.1016/S0001-8708(71)80003-8.  Google Scholar

[10]

C. T. McMullen, Dynamics on blowups of the projective plane, Publ. Math. Inst. Hautes Études Sci., 105 (2007), 49–89. doi: 10.1007/s10240-007-0004-x.  Google Scholar

[11]

The Sage Developers, SageMath, the Sage Mathematics Software System (Version 9.2), 2020., Available at https://www.sagemath.org. Google Scholar

[12]

T. Uehara, Rational surface automorphisms with positive entropy, Ann. Inst. Fourier (Grenoble), 66 (2016), 377-432.  doi: 10.5802/aif.3014.  Google Scholar

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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