J.-L. Colliot-Thélène, N. A. Karpenko, A. S. Merkur'ev, “Rational surfaces and the canonical dimension of $\mathbf{PGL}_6$”, Algebra i Analiz, 19:5 (2007), 159–178; St. Petersburg Math. J., 19:5 (2008), 793

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Algebra i Analiz, 2007, Volume 19, Issue 5, Pages 159–178 (Mi aa140)

This article is cited in 25 scientific papers (total in 25 papers)

Research Papers

Rational surfaces and the canonical dimension of

{P G L}_{6}

$\mathbf{PGL}_6$

J.-L. Colliot-Thélène^a, N. A. Karpenko^b, A. S. Merkur'ev^c

^a CNRS, Mathématiques, Université Paris-Sud, Orsay, France
^b Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie – Paris 6
^c Department of Mathematics, University of California, Los Angeles, CA, USA

Full-text PDF (227 kB) Citations (25)

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Abstract: By definition, the “canonical dimension” of an algebraic group over a field is the maximum of the canonical dimensions of the principal homogeneous spaces under that group. Over a field of characteristic zero, it is proved that the canonical dimension of the projective linear group

$\mathbf{PGL}_6$ is 3. We give two different proofs, both of which lean upon the birational classification of rational surfaces over a nonclosed field. One of the proofs involves taking a novel look at del Pezzo surfaces of degree 6.

Keywords: Algebraic group, projective linear group, rational surfaces, birational classification, canonical dimension.

Received: 29.01.2007

English version:
St. Petersburg Mathematical Journal, 2008, Volume 19, Issue 5, Pages 793–804
DOI: https://doi.org/10.1090/S1061-0022-08-01021-2

Bibliographic databases:

Document Type: Article

MSC: 14L10, 14L15

Language: Russian

Citation: J.-L. Colliot-Thélène, N. A. Karpenko, A. S. Merkur'ev, “Rational surfaces and the canonical dimension of

$\mathbf{PGL}_6$ ”, Algebra i Analiz, 19:5 (2007), 159–178; St. Petersburg Math. J., 19:5 (2008), 793–804

Citation in format AMSBIB

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\transl

\jour St. Petersburg Math. J.

\yr 2008

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\pages 793--804

\crossref{https://doi.org/10.1090/S1061-0022-08-01021-2}

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Linking options:

https://www.mathnet.ru/eng/aa140

https://www.mathnet.ru/eng/aa/v19/i5/p159

This publication is cited in the following 25 articles:

Akinari Hoshi, Hidetaka Kitayama, “Rationality Problem of Two-Dimensional Quasi-Monomial Group Actions”, Transformation Groups, 2024
Jérémy Blanc, Ivan Cheltsov, Alexander Duncan, Yuri Prokhorov, “Finite quasisimple groups acting on rationally connected threefolds”, Math. Proc. Camb. Philos. Soc., 174:3 (2023), 531–568
Andrey Trepalin, “Birational classification of pointless del Pezzo surfaces of degree 8”, Eur. J. Math., 9 (2023), 9
Kirill Zainoulline, “The Canonical Dimension of a Semisimple Group and the Unimodular Degree of a Root System”, International Mathematics Research Notices, 2023:4 (2023), 2834
Kresch A., Tschinkel Yu., “Brauer Groups of Involution Surface Bundles”, Pure Appl. Math. Q., 17:2, SI (2021), 649–669
Berg J., Varilly-Alvarado A., “Odd Order Obstructions to the Hasse Principle on General K3 Surfaces”, Math. Comput., 89:323 (2020), 1395–1416
Scavia F., “Essential Dimension and Genericity For Quiver Representations”, Doc. Math., 25 (2020), 329–364
A. Kresch, Yu. Tschinkel, “Models of Triple Covers”, Math. Notes, 105:5 (2019), 795–797
Banwait B., Fite F., Loughran D., “Del Pezzo Surfaces Over Finite Fields and Their Frobenius Traces”, Math. Proc. Camb. Philos. Soc., 167:1 (2019), 35–60
Addington N., Hassett B., Tschinkel Yu., Varilly-Alvarado A., “Cubic Fourfolds Fibered in Sextic Del Pezzo Surfaces”, Am. J. Math., 141:6 (2019), 1479–1500
Biswas I., Dhillon A., Hoffmann N., “On the Essential Dimension of Coherent Sheaves”, J. Reine Angew. Math., 735 (2018), 265–285
Xie F., “Toric Surfaces Over An Arbitrary Field Feild”, Pac. J. Math., 296:2 (2018), 481–507
Auel A., Bernardara M., “Semiorthogonal Decompositions and Birational Geometry of Del Pezzo Surfaces Over Arbitrary Fields”, Proc. London Math. Soc., 117:1 (2018), 1–64
Merkurjev A.S., “Essential Dimension”, Bull. Amer. Math. Soc., 54:4 (2017), 635–661
Christian Liedtke, Simons Symposia, Geometry Over Nonclosed Fields, 2017, 157
Merkurjev A.S., “Essential Dimension: a Survey”, Transform. Groups, 18:2 (2013), 415–481
Reichstein Z., Vistoli A., “A Genericity Theorem for Algebraic Stacks and Essential Dimension of Hypersurfaces”, J. Eur. Math. Soc., 15:6 (2013), 1999–2026
Biswas I., Dhillon A., Lemire N., “The essential dimension of stacks of parabolic vector bundles over curves”, J. K-Theory, 10:3 (2012), 455–488
Wong W., “On the essential dimension of cyclic groups”, J. Algebra, 334:1 (2011), 285–294
Brosnan P., Reichstein Z., Vistoli A., “Essential dimension of moduli of curves and other algebraic stacks”, J. Eur. Math. Soc. (JEMS), 13:4 (2011), 1079–1112

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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