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A configuration of three tangent quadrics. In this case, two of the quadrics are tangent along an entire curve
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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 |
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
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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. |
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
Figure 4.
The set $ {\rm{Conv}}(\Lambda) $
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