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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

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

[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

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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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