Two non-conjugate embeddings of $S_3\times \mathbb Z_2$ into the Cremona group II

Equivalent birational embeddings II: divisors

Two divisors in $¶^n$ are said to be Cremona equivalent if there is a Cremona modification sending one to the other. We produce infinitely many non equivalent divisorial embeddings of any variety of dimension at most 14. Then we study the special case of plane curves and rational hypersurfaces. For the latter we characterise surfaces Cremona equivalent to a plane.

On homomorphisms between Cremona groups

We look at algebraic embeddings of the Cremona group in $n$ variables $Cr_n(C)$ to the group of birational transformations $Bir(M)$ of an algebraic variety $M$. First we study geometrical properties of an example of an embedding of $Cr_2(C)$ into $Cr_5(C)$ that is due to Gizatullin. In a second part, we give a full classification of all algebraic embeddings of $Cr_2(C)$ into $Bir(M)$, where $dim(M)=3$, and generalize this result partially to algebraic embeddings of $Cr_n(C)$ into $Bir(M)$, where $dim(M)=n+1$, for arbitrary $n\geq 2$. In particular, this yields a classification of all algebraic…

Equivalent birational embedding IV: reduced varieties

Two reduced projective schemes are said to be Cremona equivalent if there is a Cremona map that maps one in the other. In this paper I revise some of the known results about Cremona equivalence and extend the main result of [MP09] to reduced schemes. This allows to prove a very general contractibility result for union of rational subvarieties.

Embeddings of SL(2,Z) into the Cremona group

Geometric and dynamic properties of embeddings of SL(2,Z) into the Cremona group are studied. Infinitely many non-conjugate embeddings which preserve the type (i.e. which send elliptic, parabolic and hyperbolic elements onto elements of the same type) are provided. The existence of infinitely many non-conjugate elliptic, parabolic and hyperbolic embeddings is also shown. In particular, a group G of automorphisms of a smooth surface S obtained by blowing-up 10 points of the complex projective plane is given. The group G is isomorphic to SL(2,Z), preserves an elliptic curve and all its elements …

Families of embeddings of the alternating group of rank 5 into the Cremona group

I study embeddings of alternating group of rank five into the Cremona group of rank three. I find all embeddings induced by A_5-del Pezzo fibrations and I study their conjugacy. As an application, I show that there is a series of continuous families of pairwise non-conjugate embeddings of alternating group of rank five into Cr$_3(\mathbb{C})$.

仿射Cremona群中加法作用的共轭性研究

On conjugacy of additive actions in the affine Cremona group

本文研究仿射Cremona群中加法作用的共轭性，证明射影空间上的任意两个加法作用可通过仿射Cremona群内由局部代数基本多项式给出的元素实现共轭，推进了代数群作用与双有理几何的相关研究。

On conjugacy classes of the Klein simple group in Cremona group

We consider countably many three dimensional $\mathtt{PSL}_2(\mathbb{F}_7)$-del Pezzo surface fibrations over $\mathbb{P}^1$. Conjecturally they are all irrational except two families, one of which is the product of a del Pezzo surface with $\mathbb{P}^1$. We show that the other model is $\mathtt{PSL}_2(\mathbb{F}_7)$-equivariantly birational to $\mathbb{P}^2\times\mathbb{P}^1$. Based on a result of Prokhorov, we show that they are non-conjugate as subgroups of the Cremona group $\mathtt{Cr}_3(\mathbb{C})$.

On solvable subgroups of the Cremona group

The Cremona group $\mathrm{Bir}(\mathbb{P}^2_\mathbb{C})$ is the group of birational self-maps of $\mathbb{P}^2_\mathbb{C}$. Using the action of $\mathrm{Bir}(\mathbb{P}^2_\mathbb{C})$ on the Picard-Manin space of $\mathbb{P}^2_\mathbb{C}$ we characterize its solvable subgroups. If $\mathrm{G}\subset\mathrm{Bir}(\mathbb{P}^2_\mathbb{C})$ is solvable, non abelian, and infinite, then up to finite index: either any element of $\mathrm{G}$ is of finite order or conjugate to an automorphism of $\mathbb{P}^2_\mathbb{C}$, or $\mathrm{G}$ preserves a unique fibration that is rational or elliptic, or $…

Some properties of the Cremona group

We recall some properties, unfortunately not all, of the Cremona group. We first begin by presenting a nice proof of the amalgamated product structure of the well-known subgroup of the Cremona group made up of the polynomial automorphisms of $\mathbb{C}^2$. Then we deal with the classification of birational maps and some applications (Tits alternative, non-simplicity...) Since any birational map can be written as a composition of quadratic birational maps up to an automorphism of the complex projective plane, we spend time on these special maps. Some questions of group theory are evoked: the c…

Centralizers of elements of infinite order in plane Cremona groups

Let $\mathbf{K}$ be an algebraically closed field. The Cremona group $\operatorname{Cr}_2(\mathbf{K})$ is the group of birational transformations of the projective plane $\mathbb{P}^2_{\mathbf{K}}$. We carry out an overall study of centralizers of elements of infinite order in $\operatorname{Cr}_2(\mathbf{K})$ which leads to a classification of embeddings of $\mathbf{Z}^2$ into $\operatorname{Cr}_2(\mathbf{K})$, as well as a classification of maximal non-torsion abelian subgroups of $\operatorname{Cr}_2(\mathbf{K})$ when $\operatorname{char}(\mathbf{K})=0$.

On elements of prime order in the plane Cremona group over a perfect field

We show that the plane Cremona group over a perfect field $k$ of characteristic $p \ge 0$ contains an element of prime order $\ell\ge 7$ not equal to $p$ if and only if there exists a 2-dimensional algebraic torus $T$ over $k$ such that $T(k)$ contains an element of order $\ell$. If $p = 0$ and $k$ does not contain a primitive $\ell$-th root of unity, we show that there are no elements of prime order $\ell > 7$ in $\Cr_2(k)$ and all elements of order 7 are conjugate.

Sofic profile and computability of Cremona groups

In this paper, we show that Cremona groups are sofic. We actually introduce a quantitative notion of soficity, called sofic profile, and show that the group of birational transformations of a d-dimensional variety has sofic profile at most polynomial of degree d. We also observe that finitely generated subgroups of the Cremona group have a solvable word problem. This provides examples of finitely generated groups with no embeddings into any Cremona group, answering a question of S. Cantat.

Finite abelian subgroups of the Cremona group of the plane

This work presents the conjugacy classes of finite abelian subgroups of the Cremona group of the plane. Using a well-known theory, this problem amounts to the study of automorphism groups of some Del Pezzo surfaces and conic bundles. We have thus to enumerate all the cases, which gives a long description, and then to show whether two cases are distinct or not, using some conjugacy invariants. For example, we use the non-rational curves fixed by one element of the group, and the action of the whole group on these curves. From this classification, we deduce a sequence of more general results on …

Five embeddings of one simple group

We propose a new method to study birational maps between Fano varieties based on multiplier ideal sheaves. Using this method, we prove equivariant birational rigidity of four Fano threefolds acted on by the group A6. As an application, we obtain that the Cremona group of rank 3 has at least five non-conjugate subgroups isomorphic to A6.

Nonrational del Pezzo fibrations admitting an action of the Klein simple group

We present a series of del Pezzo fibrations of degree $2$ admitting an action of the Klein simple group and prove their nonrationality by the reduction modulo $p$ method of Kollár. This is relevant to embeddings of the Klein simple group into the Cremona group of rank $3$.

Conjugate varieties with distinct real cohomology algebras

Using constructions of Voisin, we exhibit a smooth projective variety defined over a number field k and two complex embeddings of k, such that the two complex manifolds induced by these embeddings have non isomorphic cohomology algebras with real coefficients. This contrasts with the fact that the cohomology algebras with l-adic coefficients are canonically isomorphic for any prime number l, and answers a question of Grothendieck.

Burkhardt quartic, Barth sextic, and the icosahedron

We study two rational Fano threefolds with an action of the icosahedral group $\mathfrak{A}_5$. The first one is the famous Burkhardt quartic threefold, and the second one is the double cover of the projective space branched in the Barth sextic surface. We prove that both of them are $\mathfrak{A}_5$-Fano varieties that are $\mathfrak{A}_5$-birationally superrigid. This gives two new embeddings of the group $\mathfrak{A}_5$ into the space Cremona group.

Embeddings of groups ${\rm Aut}(F_n)$ into automorphism groups of algebraic varieties

For every positive integer $n$, we construct, using algebraic groups, an infinite family of irreducible algebraic varieties $X$,whose automorphism group ${\rm Aut}(X)$ contains the automorphism group ${\rm Aut}(F_n)$ of a free group $F_n$ of rank $n$ as a subgroup. This property implies that, for $n \geqslant 2$, such groups ${\rm Aut}(X)$ are nonamenable, and, for $n \geqslant 3$, nonlinear and contain the braid group $B_n$ on $n$ strands. Some of these varieties $X$ are affine, and among affine, some are rational and some are not, some are smooth and some are singular. As an application, we …

自由群自同构群到仿射代数簇自同构群的嵌入

Embeddings of automorphism groups of free groups into automorphism groups of affine algebraic varieties

构造了自同构群包含 Aut(F_n) 的有理仿射代数簇新无限系列，并据此改进了 Cremona 群包含 Aut(F_n) 与辫子群 B_n 的秩界。

Nonhomeomorphic conjugates of connected Shimura varieties

We show that conjugation by an automorphism of the complex numbers (as an abstract field) may change the topological fundamental group of a locally symmetric variety over C. As a consequence, we obtain a large class of algebraic varieties defined over number fields with the property that different embeddings of the number field into C give complex varieties with nonisomorphic fundamental groups.

具有模嵌入的紧致弗希斯曼群

Cocompact Fuchsian groups with a modular embedding

本文研究了具有模嵌入的紧致非算术弗希斯曼群，证明了复超曲面中完全测地曲线的模嵌入性质及算术性结论。

An arithmetic Zariski pair of line arrangements with non-isomorphic fundamental group

In a previous work, the third named author found a combinatorics of line arrangements whose realizations live in the cyclotomic group of the fifth roots of unity and such that their non-complex-conjugate embedding are not topologically equivalent in the sense that they are not embedded in the same way in the complex projective plane. That work does not imply that the complements of the arrangements are not homeomorphic. In this work we prove that the fundamental groups of the complements are not isomorphic. It provides the first example of a pair of Galois-conjugate plane curves such that the …