Plane Cremona Group Research Articles

On the conjugacy problem for finite groups in the plane Cremona group

We give a geometric condition for two finite subgroups G≃G′ of the plane Cremona group Cr2(C) to be conjugate. The argument is based on the properties of a “Burnside-type” ring Ω(P2,G) encoding rational G-surfaces.

Birational involutions of the real projective plane

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 four 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.

Borel subgroups of the plane Cremona group

Abstract It is well known that all Borel subgroups of a linear algebraic group are conjugate. Berest, Eshmatov, and Eshmatov have shown that this result also holds for the automorphism group Aut ⁢ ( 𝔸 2 ) {{\mathrm{Aut}}({\mathbb{A}}^{2})} of the affine plane. In this paper, we describe all Borel subgroups of the complex Cremona group Bir ⁢ ( ℙ 2 ) {{\rm Bir}({\mathbb{P}}^{2})} up to conjugation, proving in particular that they are not necessarily conjugate. In principle, this fact answers a question of Popov. More precisely, we prove that Bir ⁢ ( ℙ 2 ) {{\rm Bir}({\mathbb{P}}^{2})} admits Borel subgroups of any rank r ∈ { 0 , 1 , 2 } {r\in\{0,1,2\}} and that all Borel subgroups of rank r ∈ { 1 , 2 } {r\in\{1,2\}} are conjugate. In rank 0, there is a one-to-one correspondence between conjugacy classes of Borel subgroups of rank 0 and hyperelliptic curves of genus ℊ ≥ 1 {\mathcal{g}\geq 1} . Hence, the conjugacy class of a rank 0 Borel subgroup admits two invariants: a discrete one, the genus ℊ {\mathcal{g}} , and a continuous one, corresponding to the coarse moduli space of hyperelliptic curves of genus ℊ {\mathcal{g}} . This moduli space is of dimension 2 ⁢ ℊ - 1 {2\mathcal{g}-1} .

Calogero-Moser Spaces and the Invariants of Two Matrices of Degree 3

We find a minimal set of generators for the coordinate ring of Calogero-Moser space ${\boldsymbol {\mathcal {C}}}_{\boldsymbol {3}}$ and the algebraic relations among them explicitly. We give a new presentation for the algebra of 3×3 invariant matrices involving the defining relations of ${\mathbb {C}}[\boldsymbol {\mathcal {C}}_{\boldsymbol {3}}]$ . Furthermore, we find an explicit description of the GL3-invariant commuting variety of 3×3 matrices and its orbits under the action of the affine plane Cremona group.

Automorphisms of real del Pezzo surfaces and the real plane Cremona group

We study automorphism groups of real del Pezzo surfaces, concentrating on finite groups acting with invariant Picard number equal to one. As a result, we obtain a vast part of classification of finite subgroups in the real plane Cremona group.

Relations in the Cremona group over a perfect field

For perfect fields k satisfying [k ¯:k]>2, we construct new normal subgroups of the plane Cremona group and provide an elementary proof of its non-simplicity, following the melody of the recent proof by Blanc, Lamy and Zimmermann that the Cremona group of rank n over (subfields of) the complex numbers is not simple for n≥3.

The real plane Cremona group is an amalgamated product

We show that the real Cremona group of the plane is a non-trivial amalgam of two groups amalgamated along their intersection and give an alternative proof of its abelianisation.

Algebraic subgroups of the plane Cremona group over a perfect field

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.

Signature morphisms from the Cremona group over a non-closed field

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 2.

A new presentation of the plane Cremona group

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.

Infinite algebraic subgroups of the real Cremona group

We give the classification of the maximal infinite algebraic subgroups of the real Cremona group of the plane up to conjugacy and present a parametrisation space of each conjugacy class. Moreover, we show that the real plane Cremona group is not generated by a countable union of its infinite algebraic subgroups.

P-Subgroups in Automorphism Groups of Real del Pezzo Surfaces

We classify finite p-groups acting minimally on real del Pezzo surfaces. This gives a partial classification of p-subgroups in the real plane Cremona group.

On Lie problem and differential invariants for the subgroup of the plane Cremona group

Abstract We develop a new approach to the study of subgroups in the Cremona group. Namely, we consider the class of differential equations y ′ ′ = F ( x , y ) with rational right parts and its symmetry group. This group is a subgroup in the Cremona group of birational automorphisms of C 2 , which makes it possible to apply for their study methods of differential invariants and geometric theory of differential equations. Also, using algebraic methods in the theory of differential equations we obtain a global classification for such equations instead of local classifications for such problems provided by Lie, Tresse and others.

THE BOGOMOLOV-PROKHOROV INVARIANT OF SURFACES AS EQUIVARIANT COHOMOLOGY

For a complex smooth projective surface $M$ with an action of a finite cyclic group $G$ we give a uniform proof of the isomorphism between the invariant $H^1(G, H^2(M, {\mathbb Z}))$ and the first cohomology of the divisors fixed by the action, using $G$-equivariant cohomology. This generalizes the main result of Bogomolov and Prokhorov On stable conjugacy of finite subgroups of the plane Cremona group, I.

Fixed points of a finite subgroup of the plane Cremona group

We classify all finite subgroups of the plane Cremona group which have a fixed point. In other words, we determine all rational surfaces X with an action of a finite group G such that X is equivariantly birational to a surface which has a G-fixed point.

Subgroups of odd order in the real plane Cremona group

In this paper we describe conjugacy classes of finite subgroups of odd order in the group of birational automorphisms of the real projective plane.

On stable conjugacy of finite subgroups of the plane Cremona group, II

We prove that, except for a few cases, stable linearizability of finite subgroups of the plane Cremona group implies linearizability.

The conjugacy classes of finite nonsolvable subgroups in the plane Cremona group

Abstract The aim of this paper is to give a finer geometric description of the algebraic varieties parametrizing conjugacy classes of nonsolvable subgroups in the plane Cremona group

On stable conjugacy of finite subgroups of the plane Cremona group, I

AbstractWe discuss the problem of stable conjugacy of finite subgroups of Cremona groups. We compute the stable birational invariant H 1(G, Pic(X)) for cyclic groups of prime order.

Bound for the order for p-elementary subgroups in the plane cremona group over a perfect field

AbstractWe obtain a sharp bound for p-elementary subgroups in the Cremona group Cr2(k) over an arbitrary perfect field k.