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1. Algebraic subgroups of the plane Cremona group over a perfect field

Author

Julia Schneider and Susanna Zimmermann

Subjects
mathematics - algebraic geometry, 14e07, 14j50, 14l99, 20g40, Mathematics, QA1-939
Abstract

We show that any infinite algebraic subgroup of the plane Cremona group over a perfect field is contained in a maximal algebraic subgroup of the plane Cremona group. We classify the maximal groups, and their subgroups of rational points, up to conjugacy by a birational map.

Published
2021
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5. Birational involutions of the real projective plane

Author

Ivan Cheltsov, Frédéric Mangolte, Egor Yasinsky, Susanna Zimmermann, Zimmermann, Susanna, School of Mathematics - University of Edinburgh, University of Edinburgh, Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Centre de Mathématiques Laurent Schwartz (CMLS), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Angevin de Recherche en Mathématiques (LAREMA), and Université d'Angers (UA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Mathematics - Algebraic Geometry, 14E07, 14E05, 14E30, 14J45, 14P99, FOS: Mathematics, [MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG], [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], Algebraic Geometry (math.AG), [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR], [MATH.MATH-GR] Mathematics [math]/Group Theory [math.GR]
Abstract

We classify birational involutions of the real projective plane up to conjugation. In contrast with an analogous classification over the complex numbers (due to E. Bertini, G. Castelnuovo, F. Enriques, L. Bayle and A. Beauville), which includes 4 different classes of involutions, we discover 12 different classes over the reals, and provide many examples when the fixed curve of an involution does not determine its conjugacy class in the real plane Cremona group.

Published
2022

6. Quotients of higher dimensional Cremona groups

Author

Jérémy Blanc, Susanna Zimmermann, Stéphane Lamy, Mathematisches Institut Universität Basel, University of Basel (Unibas), Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Angevin de Recherche en Mathématiques (LAREMA), Université d'Angers (UA)-Centre National de la Recherche Scientifique (CNRS), ANR-18-CE40-0003,FIBALGA,Fibrations et actions de groupes algébriques(2018), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), and Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)

Subjects
Normal subgroup, Group (mathematics), General Mathematics, 010102 general mathematics, Group Theory (math.GR), Rank (differential topology), 01 natural sciences, [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR], Combinatorics, Mathematics - Algebraic Geometry, Hypersurface, Mathematics::Algebraic Geometry, Cremona group, Conic section, FOS: Mathematics, Projective space, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], 0101 mathematics, Mathematics - Group Theory, Algebraic Geometry (math.AG), Quotient, Mathematics
Abstract

We study large groups of birational transformations Bir(X), where X is a variety of dimension at least 3, defined over C or a subfield of C. Two prominent cases are when X is the projective space, in which case Bir(X) is the Cremona group of rank n, or when X is a smooth cubic hypersurface. In both cases, and more generally when X is birational to a conic bundle, we produce infinitely many distinct group homomorphisms from Bir(X) to Z/2, showing in particular that the group Bir(X) is not perfect and thus not simple. As a consequence we also obtain that the Cremona group of rank n at least 3 is not generated by linear and Jonqui\`eres elements., Comment: Item (RF4) in main definition 3.1 was modified: many thanks to Yang He for spotting the problem! Other minor changes. To appear in Acta Mathematica

Published
2021
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7. Small G-varieties

Author

Hanspeter Kraft, Andriy Regeta, Susanna Zimmermann, University of Basel (Unibas), Friedrich-Schiller-Universität = Friedrich Schiller University Jena [Jena, Germany], Laboratoire Angevin de Recherche en Mathématiques (LAREMA), Université d'Angers (UA)-Centre National de la Recherche Scientifique (CNRS), and Zimmermann, Susanna

Subjects
Mathematics - Algebraic Geometry, 14R20, 14L30, General Mathematics, [MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG], FOS: Mathematics, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], Algebraic Geometry (math.AG), [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR], [MATH.MATH-GR] Mathematics [math]/Group Theory [math.GR]
Abstract

An affine variety with an action of a semisimple group G is called “small” if every nontrivial G-orbit in X is isomorphic to the orbit of a highest weight vector. Such a variety X carries a canonical action of the multiplicative group ${\mathbb {K}^{*}}$ commuting with the G-action. We show that X is determined by the ${\mathbb {K}^{*}}$ -variety $X^U$ of fixed points under a maximal unipotent subgroup $U \subset G$ . Moreover, if X is smooth, then X is a G-vector bundle over the algebraic quotient $X /\!\!/ G$ . If G is of type ${\mathsf {A}_n}$ ( $n\geq 2$ ), ${\mathsf {C}_{n}}$ , ${\mathsf {E}_{6}}$ , ${\mathsf {E}_{7}}$ , or ${\mathsf {E}_{8}}$ , we show that all affine G-varieties up to a certain dimension are small. As a consequence, we have the following result. If $n \geq 5$ , every smooth affine $\operatorname {\mathrm {SL}}_n$ -variety of dimension $< 2n-2$ is an $\operatorname {\mathrm {SL}}_n$ -vector bundle over the smooth quotient $X /\!\!/ \operatorname {\mathrm {SL}}_n$ , with fiber isomorphic to the natural representation or its dual.

Published
2020

8. The decomposition group of a line in the plane

Author

Isac Hedén, Susanna Zimmermann, and University of Basel (Unibas)

Subjects
Pure mathematics, Degree (graph theory), Plane (geometry), Group (mathematics), Applied Mathematics, General Mathematics, 010102 general mathematics, 01 natural sciences, Decomposition, [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR], Mathematics - Algebraic Geometry, Mathematics::Group Theory, Mathematics::Algebraic Geometry, Product (mathematics), 0103 physical sciences, Line (geometry), FOS: Mathematics, 14E07, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], 010307 mathematical physics, 0101 mathematics, Element (category theory), Algebraic Geometry (math.AG), Mathematics
Abstract

We show that the decomposition group of a line $L$ in the plane, i.e. the subgroup of plane birational transformations that send $L$ to itself birationally, is generated by its elements of degree 1 and one element of degree 2, and that it does not decompose as a non-trivial amalgamated product., 15 pages, 7 figures

Published
2017
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9. Signature morphisms from the Cremona group over a non-closed field

Author

Susanna Zimmermann, Stéphane Lamy, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), ANR-18-CE40-0003,FIBALGA,Fibrations et actions de groupes algébriques(2018), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), and Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)

Subjects
Pure mathematics, General Mathematics, Field (mathematics), Group Theory (math.GR), 01 natural sciences, [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR], Surjective function, Mathematics - Algebraic Geometry, Mathematics::Group Theory, Morphism, Mathematics::Algebraic Geometry, 0103 physical sciences, FOS: Mathematics, Order (group theory), Galois extension, 0101 mathematics, Algebraic Geometry (math.AG), Mathematics, Applied Mathematics, 010102 general mathematics, Free product, Cremona group, Perfect field, 010307 mathematical physics, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], Mathematics - Group Theory
Abstract

We prove that the plane Cremona group over a perfect field with at least one Galois extension of degree 8 is a non-trivial amalgam, and that it admits a surjective morphism to a free product of groups of order two., Comment: Statements only over perfect fields instead of arbitrary fields (see Remark 1.3). A few other minor corrections

Published
2020
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10. Bijective Cremona transformations of the plane

Author

Shamil Asgarli, Kuan-Wen Lai, Masahiro Nakahara, and Susanna Zimmermann

Subjects
Mathematics - Algebraic Geometry, General Mathematics, FOS: Mathematics, General Physics and Astronomy, 14E07 (Primary) 14G15, 20B30 (Secondary), Algebraic Geometry (math.AG)
Abstract

We study the birational self-maps of the projective plane over finite fields that induce permutations on the set of rational points. As a main result, we prove that no odd permutation arises over a non-prime finite field of characteristic two, which completes the investigation initiated by Cantat about which permutations can be realized this way. Main ingredients in our proof include the invariance of parity under groupoid conjugations by birational maps, and a list of generators for the group of such maps., 51 pages, removed section 5.A and made improvements on exposition, to appear in Selecta Mathematica

Published
2019

11. Continuous automorphisms of Cremona groups

Author

Christian Urech, Susanna Zimmermann, Ecole Polytechnique Fédérale de Lausanne (EPFL), Laboratoire Angevin de Recherche en Mathématiques (LAREMA), Université d'Angers (UA)-Centre National de la Recherche Scientifique (CNRS), Swiss National Science Foundation project P2BSP2_175008Project Etoiles montantes of the Région Pays de la LoireProject PEPS 2019 'JC/JC', and ANR-18-CE40-0003,FIBALGA,Fibrations et actions de groupes algébriques(2018)

Subjects
Pure mathematics, General Mathematics, automorphisms of groups, Field (mathematics), Group Theory (math.GR), Rank (differential topology), 01 natural sciences, [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR], cremona groups, Mathematics::Group Theory, Mathematics - Algebraic Geometry, Mathematics::Algebraic Geometry, 0103 physical sciences, Euclidean topology, FOS: Mathematics, 0101 mathematics, Algebraic Geometry (math.AG), Mathematics, Mathematics::Commutative Algebra, Group (mathematics), 010102 general mathematics, Mathematics::History and Overview, 2010 MSC: 20F28, 14E07, 16. Peace & justice, Automorphism, Homeomorphism, continuous maps, Cremona group, 010307 mathematical physics, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], Mathematics - Group Theory
Abstract

International audience; We show that if a group automorphism of a Cremona group of arbitrary rank is also a homeomorphism with respect to either the Zariski or the Euclidean topology, then it is inner up to a field automorphism of the base-field. Moreover, we show that a similar result holds if we consider groups of polynomial automorphisms of affine spaces instead of Cremona groups.

Published
2019
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12. A new presentation of the plane Cremona group

Author

Christian Urech, Susanna Zimmermann, Imperial College London, Laboratoire Angevin de Recherche en Mathématiques (LAREMA), and Université d'Angers (UA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Pure mathematics, Mathematics::Commutative Algebra, Plane (geometry), Applied Mathematics, General Mathematics, 010102 general mathematics, 01 natural sciences, [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR], symbols.namesake, Mathematics - Algebraic Geometry, Mathematics::Algebraic Geometry, Cremona group, Simple (abstract algebra), 0103 physical sciences, symbols, FOS: Mathematics, 010307 mathematical physics, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], 0101 mathematics, Algebraically closed field, Presentation (obstetrics), Noether's theorem, Algebraic Geometry (math.AG), Mathematics
Abstract

International audience; We give a presentation of the plane Cremona group over an algebraically closed field with respect to the generators given by the Theorem of Noether and Castelnuovo. This presentation is particularly simple and can be used for explicit calculations.

Published
2018

13. TOPOLOGICAL SIMPLICITY OF THE CREMONA GROUPS

Author

Susanna Zimmermann, Jérémy Blanc, and Zimmermann, Susanna

Subjects
Pure mathematics, General Mathematics, media_common.quotation_subject, Dimension (graph theory), 01 natural sciences, Mathematics - Algebraic Geometry, Mathematics::Algebraic Geometry, Simple (abstract algebra), 0103 physical sciences, Euclidean topology, Simplicity, 0101 mathematics, 14E07, 22F50, 14R20, Mathematics, media_common, Mathematics - General Topology, [MATH.MATH-GR] Mathematics [math]/Group Theory [math.GR], Mathematics::Commutative Algebra, Group (mathematics), 010102 general mathematics, [MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG], Cremona group, Line (geometry), 010307 mathematical physics, Affine transformation, Mathematics - Group Theory
Abstract

The Cremona group is topologically simple when endowed with the Zariski or Euclidean topology, in any dimension $\ge 2$ and over any infinite field. Two elements are moreover always connected by an affine line, so the group is path-connected.

Published
2018

14. The decomposition groups of plane conics and plane rational cubics

Author

Tom Ducat, Susanna Zimmermann, Isac Hedén, Research Institute for Mathematical Sciences (RIMS), Kyoto University [Kyoto], Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Kyoto University, Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Degree (graph theory), Plane curve, Plane (geometry), Group (mathematics), General Mathematics, 010102 general mathematics, 01 natural sciences, Cubic plane curve, [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR], Combinatorics, Mathematics - Algebraic Geometry, Mathematics::Algebraic Geometry, Conic section, FOS: Mathematics, 14E07, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], 0101 mathematics, Algebraic Geometry (math.AG), Mathematics
Abstract

The decomposition group of an irreducible plane curve $X\subset\mathbb P^2$ is the subgroup $\mathrm{Dec}(X)\subset\mathrm{Bir}(\mathbb P^2)$ of birational maps which restrict to a birational map of $X$. We show that $\mathrm{Dec}(X)$ is generated by its elements of degree $\leq2$ when $X$ is either a conic or rational cubic curve., Comment: 12 pages, 6 figures. Comments are welcome

Published
2017
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15. The Abelianisation of the real Cremona group

Author

Susanna Zimmermann, University of Basel (Unibas), Laboratoire Angevin de Recherche en Mathématiques (LAREMA), and Université d'Angers (UA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
General Mathematics, 010102 general mathematics, Cremona group, 01 natural sciences, [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR], birational geometry, Algebra, Mathematics::Group Theory, Mathematics - Algebraic Geometry, Mathematics::Algebraic Geometry, 0103 physical sciences, FOS: Mathematics, real algebraic geometry, 14E07, 14P99, 14E07, 010307 mathematical physics, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], 0101 mathematics, Algebraic Geometry (math.AG), 14P99, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract

We present the Abelianisation of the birational transformations of $\mathbb{P}_{\mathbb{R}}^2$. Its kernel is equal to the normal subgroup generated by $PGL_3(\mathbb{R})$, and contains all elements of degree $\le 4$. The description of the quotient yields the existence of normal subgroups of index $2^n$ for any n and that the normal subgroup generated by any countable set of elements is a proper subgroup.

Published
2015
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16. Algebraic subgroups of the plane Cremona group over a perfect field

Author

Julia Schneider, Susanna Zimmermann, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Laboratoire Angevin de Recherche en Mathématiques (LAREMA), Université d'Angers (UA)-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Pure mathematics, Algebra and Number Theory, Plane (geometry), [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR], 14E07, 14J50, 14L99, 20G40, Mathematics - Algebraic Geometry, Mathematics::Group Theory, Mathematics::Algebraic Geometry, Cremona group, FOS: Mathematics, Perfect field, Geometry and Topology, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], Algebraic number, Algebraic Geometry (math.AG), ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract

We show that any infinite algebraic subgroup of the plane Cremona group over a perfect field is contained in a maximal algebraic subgroup of the plane Cremona group. We classify the maximal groups, and their subgroups of rational points, up to conjugacy by a birational map.

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