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Julie Déserti

Publications and source records attributed to Julie Déserti.

At least 19 recordsLinked to original sources
arXiv2023

Algebraic properties of the group of germs of diffeomorphisms

We establish some algebraic properties of the group $\mathrm{Diff}(\mathbb{C}^n,0)$ of germs of analytic diffeomorphisms of $\mathbb{C}^n$, and its formal completion $\widehat{\mathrm{Diff}}(\mathbb{C}^n,0)$. For instance we describe the commutator of $\mathrm{Diff}(\mathbb{C}^n,0)$, but also prove that any finitely generated subgroup of $\mathrm{Diff}(\mathbb{C}^n,0)$ is residually finite; we thus obtain some constraints of groups that embed into $\mathrm{Diff}(\mathbb{C}^n,0)$. We show that $\widehat{\mathrm{Diff}}(\mathbb{C}^n,0)$ is an Hopfian group, and that $\mathrm{Diff}(\mathbb{C}^n,0)$ and $\widehat{\mathrm{Diff}}(\mathbb{C}^n,0)$ are not co-Hopfian. We end by the description of the automorphisms groups of $\widehat{\mathrm{Diff}}(\mathbb{C},0)$, and $\mathrm{Diff}(\mathbb{C},0)$.

math.GR↗
arXiv2023

Singularities of holomorphic codimension one foliations of the complex projective plane

We prove that any holomorphic codimension 1 foliation on the complex projective plane has at most one singular point up to the action of an ad-hoc birational self map of the complex projective plane into itself. Consequently, any algebraic foliation on the affine plane has no singularities up to the action of a suitable birational self map of the complex projective plane into itself.

math.DS↗
arXiv2021

Three chapters on Cremona groups

In the first part of this article, we answer a question of I. Dolgachev, which is related to the following problem: given a birational map $f\in\mathrm{Bir}(\mathbb{P}^m_\mathbf{k})$ and a linear projective map $A\in\mathrm{PGL}_{m+1}(\mathbf{k})$, when is $A\circ f$ regularizable? Dolgachev's initial question is whether this may happen for all $A$ in $\mathrm{PGL}_{m+1}(\mathbf{k})$, and the answer is negative. We then look at the sequence $n\mapsto\mathrm{deg}\, f^n$, $f\in\mathrm{Bir}(\mathbb{P}^2_\mathbf{k})$. We show that there is no constraint on the sequence $n\mapsto\mathrm{deg}\, f^n-\mathrm{deg}\, f^{n-1}$ for small values of $n$. Finally we study the degree of pencils of curves which are invariant by a birational map. When f is a Halphen or Jonquières twist, we prove that this degree is bounded by a function of $\mathrm{deg}\, f$. We derive corollaries on the structure of conjugacy classes, and their properties with respect to the Zariski topology of $\mathrm{Bir}(\mathbb{P}^2_\mathbf{k})$.

math.AG↗
arXiv2021

On regularizable birational maps

Bedford asked if there exists a birational self map $f$ of the complex projective plane such that for any automorphism $A$ of the complex projective plane $A\circ f$ is not conjugate to an automorphism. Blanc gave such a $f$ of degree $6$ and asked if there exists an example of smaller degree. In this article we give an example of degree $5$.

math.DS↗
arXiv2017

Cremona maps and involutions

We deal with the following question of Dolgachev : is the Cremona group generated by involutions ? Answer is yes in dimension $2$ (Cerveau-Deserti). We give an upper bound of the minimal number $\mathfrak{n}_φ$ of involutions we need to write a birational self map $φ$ of $\mathbb{P}^2_\mathbb{C}$. We prove that de Jonquières maps of $\mathbb{P}^3_\mathbb{C}$ and maps of small bidegree of $\mathbb{P}^3_\mathbb{C}$ can be written as a composition of involutions of $\mathbb{P}^3_\mathbb{C}$ and give an upper bound of $\mathfrak{n}_φ$ for such maps $φ$. We get similar results in particular for automorphisms of $(\mathbb{P}^1_\mathbb{C})^n$, automorphisms of $\mathbb{P}^n_\mathbb{C}$, tame automorphisms of $\mathbb{C}^n$, monomial maps of $\mathbb{P}^n_\mathbb{C}$, and elements of the subgroup generated by the standard involution of $\mathbb{P}^n_\mathbb{C}$ and $\mathrm{PGL}(n+1,\mathbb{C})$.

math.AG↗
arXiv2017

Quarto-quartic birational maps of $\mathbb{P}_3(\mathbb{C})$

We construct a determinantal family of quarto-quartic transformations of a complex projective space of dimension $3$ from trigonal curves of degree $8$ and genus $5$. Moreover we show that the variety of $(4,4)$-birational maps of $\mathbb{P}_3(\mathbb{C})$ has at least four irreducible components and describe three of them.

math.AG↗
arXiv2016

Degree growth of polynomial automorphisms and birational maps: some examples

We provide the existence of new degree growths in the context of polynomial automorphisms of $\mathbb{C}^k$: if $k$ is an integer $\geq 3$, then for any $\ell\leq \left[\frac{k-1}{2}\right]$ there exist polynomial automorphisms $f$ of $\mathbb{C}^k$ such that $\mathrm{deg}\, f^n\sim n^\ell$. We also give counter-examples in dimension $k\geq 3$ to some classical properties satisfied by polynomial automorphisms of $\mathbb{C}^2$. We provide the existence of new degree growths in the context of birational maps of $\mathbb{P}^k_\mathbb{C}$: assume $k\geq 3$; forall $0\leq\ell\leq k$ there exist birational maps $ϕ$ of $\mathbb{P}^k_\mathbb{C}$ such that $\mathrm{deg}\, ϕ^n\sim n^\ell$.

math.DS↗
arXiv2016

Dynamics of a family of polynomial automorphisms of $\mathbb{C}^3$, a phase transition

The polynomial automorphisms of the affine plane have been studied a lot: if $f$ is such an automorphism, then either $f$ preserves a rational fibration, has an uncountable centralizer and its first dynamical degree equals $1$, or $f$ preserves no rational curves, has a countable centralizer and its first dynamical degree is $>1$. In higher dimensions there is no such description. In this article we study a family $(Ψ_α)_α$ of polynomial automorphisms of $\mathbb{C}^3$. We show that the first dynamical degree of $Ψ_α$ is $>1$, that $Ψ_α$ preserves a unique rational fibration and has an uncountable centralizer. We then describe the dynamics of the family $(Ψ_α)_α$, in particular the speed of points escaping to infinity. We also observe different behaviors according to the value of the parameter $α$.

math.DS↗
arXiv2016

Jonquières maps and $\mathrm{SL}(2;\mathbb{C})$-cocycles

We start the study of the family of birational maps $(f_{α,β})$ of $\mathbb{P}^2_\mathbb{C}$ in \cite{Deserti}. For generic $α$ and $β$ of modulus 1 the centraliser of $f_{α,β}$ is trivial, the topological entropy of $f_{α,β}$ is 0, there exist two areas of linearisation: in the first one the closure of the orbit of a point is a torus, in the other one the closure of the orbit of a point is the union of two circles. On $\mathbb{P}^1_\mathbb{C}\times \mathbb{P}^1_\mathbb{C}$ any $f_{α,β}$ can be viewed as a cocyle; using recent results about $\mathrm{SL}(2;\mathbb{C})$-cocycles (\cite{Avila}) we determine the \textsc{Lyapunov} exponent of the cocyle associated to $f_{α,β}$.

math.DS↗
arXiv2016

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 $\mathrm{G}$ is, up to conjugacy, a subgroup of the group generated by one hyperbolic monomial map and the diagonal automorphisms. We also give some corollaries.

math.AG↗
arXiv2015

Some properties of the group of birational maps generated by the automorphisms of $\mathbb{P}^n_\mathbb{C}$ and the standard involution

We give some properties of the subgroup $G_n(\mathbb{C})$ of the group of birational self-maps of $\mathbb{P}^n_\mathbb{C}$ generated by the standard involution and the group of automorphisms of $\mathbb{P}^n_\mathbb{C}$. We prove that there is no nontrivial finite-dimensional linear representation of $G_n(\mathbb{C})$. We also establish that $G_n(\mathbb{C})$ is perfect, and that $G_n(\mathbb{C})$ equipped with the Zariski topology is simple. Furthermore if $φ$ is an automorphism of $\mathrm{Bir}(\mathbb{P}^n_\mathbb{C})$, then up to birational conjugacy, and up to the action of a field automorphism $φ_{\vert G_n(\mathbb{C})}$ is trivial.

math.AG↗