Automorphisms of surfaces over fields of positive characteristic

We study automorphism and birational automorphism groups of varieties over fields of positive characteristic from the point of view of Jordan and p-Jordan property. In particular, we show that the Cremona group of rank 2 over a field of characteristic p > 0 is p-Jordan, and the birational automorphism group of an arbitrary geometrically irreducible algebraic surface is nilpotently p-Jordan of class at most 2. Also, we show that the automorphism group of a smooth geometrically irreducible projective variety of non-negative Kodaira dimension is Jordan in the usual sense.

[1]  V. Popov.  ON THE MAKAR-LIMANOV, DERKSEN INVARIANTS, AND FINITE AUTOMORPHISM GROUPS OF ALGEBRAIC VARIETIES , 2010, 1001.1311.

[2]  M. Hanamura.  Structure of birational automorphism groups, I: non-uniruled varieties , 1988 .

[3]  I. Dolgachev,et al.  Numerically trivial automorphisms of Enriques surfaces in characteristic 2 , 2017, Journal of the Mathematical Society of Japan.

[4]  Egor Yasinsky,et al.  The Jordan constant for Cremona group of rank 2 , 2016, 1610.09654.

[5]  C. Shramov,et al.  Jordan property for groups of birational selfmaps , 2013, Compositio Mathematica.

[6]  H. Grimmer,et al.  Comment on a paper by Tahara on the finite subgroups of $GL(3,Z)$ , 1972 .

[7]  M. Martin-Deschamps,et al.  Applications rationnelles séparables dominantes sur une variété de type général , 1978 .

[8]  Yi Gu.  Surface fibrations with large equivariant automorphism group , 2020, 2004.12656.

[9]  I. Dolgachev,et al.  Automorphisms of cubic surfaces in positive characteristic , 2017, Izvestiya: Mathematics.

[10]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[11]  T. Willmore.  Algebraic Geometry , 1973, Nature.

[12]  Serge Lang,et al.  Abelian varieties , 1983 .

[13]  M. Brion.  Some structure theorems for algebraic groups , 2015, 1509.03059.

[14]  W. Feit,et al.  An Analogue of Jordan's Theorem in Characteristic p , 1966 .

[15]  飯高 茂.  Algebraic geometry : an introduction to birational geometry of algebraic varieties , 1982 .

[16]  B. Green,et al.  Approximate groups, III: the unitary case , 2010, 1006.5160.

[17]  C. Wampler,et al.  Basic Algebraic Geometry , 2005 .

[18]  A. Ogus.  Supersingular K3 crystals , 2019 .

[19]  J. Keum.  Orders of automorphisms of K3 surfaces , 2012, 1203.5616.

[20]  C. Liedtke.  Algebraic Surfaces in Positive Characteristic , 2009, 0912.4291.

[21]  I. M. I. Riera.  ALMOST FIXED POINTS OF FINITE GROUP ACTIONS ON MANIFOLDS WITHOUT ODD COHOMOLOGY , 2013, Transformation Groups.

[22]  Yuri Prokhorov,et al.  Jordan property for Cremona groups , 2012, 1211.3563.

[23]  C. Birkar,et al.  Existence of Mori fibre spaces for 3-folds in char $p$ , 2014, 1410.4511.

[24]  C. Shramov,et al.  Automorphism Groups of Compact Complex Surfaces , 2017, International Mathematics Research Notices.

[25]  E. Breuillard.  An exposition of Jordan’s original proof of his theorem on finite subgroups of GLn(ℂ) , 2023, Model Theory.

[26]  C. Birkar.  Existence of flips and minimal models for 3-folds in char p , 2013, 1311.3098.

[27]  H. Minkowski,et al.  Zur Theorie der positiven quadratischen Formen. , 1887 .

[28]  S. Kleiman.  Toward a numerical theory of ampleness , 1966 .

[29]  V A Iskovskih,et al.  Minimal Models of Rational Surfaces Over Arbitrary Fields , 1980 .

[30]  A. Grothendieck,et al.  Éléments de géométrie algébrique , 1960 .

[31]  W. L. Baily,et al.  Enriques ' Classification of Surfaces in Char . p , II , 2005 .

[32]  Y. Tschinkel,et al.  Birational geometry, rational curves, and arithmetic , 2013 .

[33]  I. M. I. Riera.  Jordan's theorem for the diffeomorphism group of some manifolds , 2010 .

[34]  S. Kleiman.  Les Theoremes de Finitude pour le Foncteur de Picard , 1971 .

[35]  K. Ueno.  Classification theory of algebraic varieties and compact complex spaces , 1975 .

[36]  I. Dolgachev,et al.  Finite Subgroups of the Plane Cremona Group , 2006, math/0610595.

[37]  Joseph Lipman,et al.  Desingularization of two-dimensional schemes , 1978 .

[38]  Automorphism schemes of (quasi-)bielliptic surfaces , 2020, 2008.11679.

[39]  Jean-Philippe Furter,et al.  Topologies and structures of the Cremona groups , 2012, 1210.6960.

[40]  János Kollár,et al.  Rational curves on algebraic varieties , 1995, Ergebnisse der Mathematik und ihrer Grenzgebiete.

[41]  I. Dolgachev,et al.  On Elements of Prime Order in the Plane Cremona Group over a Perfect Field , 2007, 0707.4305.

[42]  Fei Hu.  Jordan property for algebraic groups and automorphism groups of projective varieties in arbitrary characteristic , 2018, 1804.10946.

[43]  Carlisle S. H. King.  Generation of finite simple groups by an involution and an element of prime order , 2016, 1603.04717.

[44]  L. Roth,et al.  Algebraic Surfaces , 1950, Nature.

[45]  Mohammad Reza Darafsheh,et al.  Order of elements in the groups related to the general linear group , 2005, Finite Fields Their Appl..

[46]  小平 邦彦,et al.  Global analysis : papers in honor of K. Kodaira , 2015 .

[47]  V. Popov.  Birational splitting and algebraic group actions , 2015, 1502.02167.

[48]  C. Shramov,et al.  Jordan constant for Cremona group of rank 3 , 2016, 1608.00709.

[49]  C. Birkar.  Singularities of linear systems and boundedness of Fano varieties , 2016, Annals of Mathematics.

[50]  M. Hanamura.  On the birational automorphism groups of algebraic varieties , 1987 .

[51]  A. Weil,et al.  On Algebraic Groups of Transformations , 1955 .

[52]  G. Ballew,et al.  The Arithmetic of Elliptic Curves , 2020, Elliptic Curves.

[53]  De-Qi Zhang,et al.  Jordan property for non-linear algebraic groups and projective varieties , 2015, 1507.02230.

[54]  I︠u︡. I. Manin,et al.  Cubic forms; algebra, geometry, arithmetic , 1974 .

[55]  Y. Zarhin.  Theta Groups and Products of Abelian and Rational Varieties , 2010, Proceedings of the Edinburgh Mathematical Society.

[56]  J. Lipman.  Rational singularities, with applications to algebraic surfaces and unique factorization , 1969 .

[57]  R. Lazarsfeld.  Classical setting : line bundles and linear series , 2004 .

[58]  Qing Liu,et al.  Algebraic Geometry and Arithmetic Curves , 2002 .

[59]  B. Fantechi.  Fundamental Algebraic Geometry , 2005 .

[60]  Maureen T. Carroll.  Geometry , 2017, MAlkahtani Mathematics.

[61]  D. Correia,et al.  76 , 2018, The Devil's Fork.

[62]  Jean-Pierre Serre.  Algebraic Groups and Class Fields , 1987 .

[63]  David Schwein,et al.  Étale Cohomology , 2018, Translations of Mathematical Monographs.

[64]  C. Shramov,et al.  Bounded automorphism groups of compact complex surfaces , 2019, Sbornik: Mathematics.

[65]  C. Curtis,et al.  Representation theory of finite groups and associated algebras , 1962 .

[66]  Walter Feit,et al.  Finite Simple Groups , 1983 .

[67]  Michael Larsen,et al.  Finite subgroups of algebraic groups , 2011 .

[68]  T. Bandman,et al.  JORDAN GROUPS AND ALGEBRAIC SURFACES , 2014, 1404.1581.

[69]  Jean-Pierre Serre,et al.  Bounds for the orders of the finite subgroups of G(k) , 2010, 1011.0346.

[70]  Jean-Pierre Serre.  A Minkowski-style bound for the order of the finite subgroups of the Cremona group of rank 2 over an arbitrary field , 2009, 0903.0514.

[71]  W. Waterhouse.  Two generators for the general linear groups over finite fields , 1989 .

[72]  Christopher D. Hacon,et al.  On the three dimensional minimal model program in positive characteristic , 2013, 1302.0298.

[73]  Dan Segal,et al.  Finite Group Theory , 2003 .