# American Institute of Mathematical Sciences

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

## Tri-Coble surfaces and their automorphisms

 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.

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. [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. [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. [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. [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. [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. [7] I. V. Dolgachev, Classical Algebraic Geometry: A Modern View, Cambridge University Press, Cambridge, 2012.  doi: 10.1017/CBO9781139084437. [8] J. Lesieutre, Code for "{T}ri-{C}oble surfaces and their automorphisms'', ScholarSphere, Penn State University Libraries, 2021. doi: 10.26207/xwyq-hc68. [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. [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. [11] The Sage Developers, SageMath, the Sage Mathematics Software System (Version 9.2), 2020., Available at https://www.sagemath.org. [12] T. Uehara, Rational surface automorphisms with positive entropy, Ann. Inst. Fourier (Grenoble), 66 (2016), 377-432.  doi: 10.5802/aif.3014.

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. [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. [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. [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. [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. [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. [7] I. V. Dolgachev, Classical Algebraic Geometry: A Modern View, Cambridge University Press, Cambridge, 2012.  doi: 10.1017/CBO9781139084437. [8] J. Lesieutre, Code for "{T}ri-{C}oble surfaces and their automorphisms'', ScholarSphere, Penn State University Libraries, 2021. doi: 10.26207/xwyq-hc68. [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. [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. [11] The Sage Developers, SageMath, the Sage Mathematics Software System (Version 9.2), 2020., Available at https://www.sagemath.org. [12] T. Uehara, Rational surface automorphisms with positive entropy, Ann. Inst. Fourier (Grenoble), 66 (2016), 377-432.  doi: 10.5802/aif.3014.
A configuration of three tangent quadrics. In this case, two of the quadrics are tangent along an entire curve
$\tau_ \mathbf p$ 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
The action of $G$ on $\Delta$
The set ${\rm{Conv}}(\Lambda)$
 [1] B. Harbourne, P. Pokora, H. Tutaj-Gasińska. On integral Zariski decompositions of pseudoeffective divisors on algebraic surfaces. Electronic Research Announcements, 2015, 22: 103-108. doi: 10.3934/era.2015.22.103 [2] Piotr Pokora, Tomasz Szemberg. Minkowski bases on algebraic surfaces with rational polyhedral pseudo-effective cone. Electronic Research Announcements, 2014, 21: 126-131. doi: 10.3934/era.2014.21.126 [3] Alfonso Artigue. Expansive flows of surfaces. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 505-525. doi: 10.3934/dcds.2013.33.505 [4] Kariane Calta, John Smillie. Algebraically periodic translation surfaces. Journal of Modern Dynamics, 2008, 2 (2) : 209-248. doi: 10.3934/jmd.2008.2.209 [5] Anton Petrunin. Correction to: Metric minimizing surfaces. Electronic Research Announcements, 2018, 25: 96-96. doi: 10.3934/era.2018.25.010 [6] Anton Petrunin. Metric minimizing surfaces. Electronic Research Announcements, 1999, 5: 47-54. [7] Siran Li, Jiahong Wu, Kun Zhao. On the degenerate boussinesq equations on surfaces. Journal of Geometric Mechanics, 2020, 12 (1) : 107-140. doi: 10.3934/jgm.2020006 [8] Yong Lin, Gábor Lippner, Dan Mangoubi, Shing-Tung Yau. Nodal geometry of graphs on surfaces. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 1291-1298. doi: 10.3934/dcds.2010.28.1291 [9] Gabriele Beltramo, Primoz Skraba, Rayna Andreeva, Rik Sarkar, Ylenia Giarratano, Miguel O. Bernabeu. Euler characteristic surfaces. Foundations of Data Science, 2021  doi: 10.3934/fods.2021027 [10] Eduard Duryev, Charles Fougeron, Selim Ghazouani. Dilation surfaces and their Veech groups. Journal of Modern Dynamics, 2019, 14: 121-151. doi: 10.3934/jmd.2019005 [11] Alexandre Girouard, Iosif Polterovich. Upper bounds for Steklov eigenvalues on surfaces. Electronic Research Announcements, 2012, 19: 77-85. doi: 10.3934/era.2012.19.77 [12] Seung Won Kim, P. Christopher Staecker. Dynamics of random selfmaps of surfaces with boundary. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 599-611. doi: 10.3934/dcds.2014.34.599 [13] Roberto Paroni, Podio-Guidugli Paolo, Brian Seguin. On the nonlocal curvatures of surfaces with or without boundary. Communications on Pure and Applied Analysis, 2018, 17 (2) : 709-727. doi: 10.3934/cpaa.2018037 [14] José Ginés Espín Buendía, Daniel Peralta-salas, Gabriel Soler López. Existence of minimal flows on nonorientable surfaces. Discrete and Continuous Dynamical Systems, 2017, 37 (8) : 4191-4211. doi: 10.3934/dcds.2017178 [15] Gareth Ainsworth. The attenuated magnetic ray transform on surfaces. Inverse Problems and Imaging, 2013, 7 (1) : 27-46. doi: 10.3934/ipi.2013.7.27 [16] François Béguin. Smale diffeomorphisms of surfaces: a classification algorithm. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 261-310. doi: 10.3934/dcds.2004.11.261 [17] Robert S. Strichartz. Average error for spectral asymptotics on surfaces. Communications on Pure and Applied Analysis, 2016, 15 (1) : 9-39. doi: 10.3934/cpaa.2016.15.9 [18] Gareth Ainsworth. The magnetic ray transform on Anosov surfaces. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 1801-1816. doi: 10.3934/dcds.2015.35.1801 [19] Leo T. Butler. A note on integrable mechanical systems on surfaces. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1873-1878. doi: 10.3934/dcds.2014.34.1873 [20] Martin Bauer, Philipp Harms, Peter W. Michor. Sobolev metrics on shape space of surfaces. Journal of Geometric Mechanics, 2011, 3 (4) : 389-438. doi: 10.3934/jgm.2011.3.389

2020 Impact Factor: 0.848