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Pierre-Marie Poloni

Publications and source records attributed to Pierre-Marie Poloni.

18 recordsLinked to original sources
arXiv2023

Complements of hypersurfaces in projective spaces

We study the complement problem in projective spaces $\mathbb{P}^n$ over any algebraically closed field: If $H, H' \subseteq \mathbb{P}^n$ are irreducible hypersurfaces of degree $d$ such that the complements $\mathbb{P}^n \setminus H$, $\mathbb{P}^n \setminus H'$ are isomorphic, are the hypersurfaces $H$, $H'$ isomorphic? For $n = 2$, the answer is positive if $d\leq 7$ and there are counterexamples when $d = 8$. In contrast we provide counterexamples for all $n, d \geq 3$ with $(n, d) \neq (3, 3)$. Moreover, we show that the complement problem has an affirmative answer for $d = 2$ and give partial results in case $(n, d) = (3, 3)$. In the course of the exposition, we prove that rational normal projective surfaces admitting a desingularisation by trees of smooth rational curves are piecewise isomorphic if and only if they coincide in the Grothendieck ring, answering affirmatively a question posed by Larsen and Lunts for such surfaces.

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arXiv2021

Real forms of some Gizatullin surfaces and Koras-Russell threefolds

We describe the real forms of Gizatullin surfaces of the form $xy=p(z)$ and of Koras-Russell threefolds of the first kind. The former admit zero, two, three, four or six isomorphism classes of real forms, depending on the degree and the symmetries of the polynomial~$p$. The latter, which are threefolds given by an equation of the form $x^dy+z^k+x+t^\ell=0$, all admit exactly one real form up to isomorphism.

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arXiv2020

Bivariables and Vénéreau polynomials

We study a family of polynomials introduced by Daigle and Freudenburg, which contains the famous Vénéreau polynomials and defines $\mathbb{A}^2$-fibrations over $\mathbb{A}^2$. According to the Dolgachev-Weisfeiler conjecture, every such fibration should have the structure of a locally trivial $\mathbb{A}^2$-bundle over $\mathbb{A}^2$. We follow an idea of Kaliman and Zaidenberg to show that these fibrations are locally trivial $\mathbb{A}^2$-bundles over the punctured plane, all of the same specific form $X_f$, depending on an element $f\in k[a^{\pm 1},b^{\pm 1}][x]$. We then introduce the notion of bivariables and show that the set of bivariables is in bijection with the set of locally trivial bundles $X_f$ that are trivial. This allows us to give another proof of Lewis's result stating that the second Vénéreau polynomial is a variable and also to trivialise other elements of the family $X_f$. We hope that the terminology and methods developed here may lead to future study of the whole family $X_f$.

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arXiv2020

Isomorphisms between cylinders over Danielewski surfaces

A special Danielewski surface is an affine surface which is the total space of a principal $(\mathbb{C},+)$-bundle over an affine line with a multiple origin. Using a fiber product trick introduced by Danielewski, it is known that cylinders over two such surfaces are always isomorphic provided that both bases have the same number of origins. The goal of this note is to give an explicit method to find isomorphisms between cylinders over special Danielewski surfaces. The method is based on the construction of appropriate locally nilpotent derivations.

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arXiv2017

On the maximality of the triangular subgroup

We prove that the subgroup of triangular automorphisms of the complex affine $n$-space is maximal among all solvable subgroups of $\mathrm{Aut}(\mathbb{A}_{\mathbb{C}}^n)$ for every $n$. In particular, it is a Borel subgroup of $\mathrm{Aut}(\mathbb{A}_{\mathbb{C}}^n)$, when the latter is viewed as an ind-group. In dimension two, we prove that the triangular subgroup is a maximal closed subgroup. Nevertheless, it is not maximal among all subgroups of $\mathrm{Aut}(\mathbb{A}_{\mathbb{C}}^2)$. Given an automorphism $f$ of $\mathbb{A}_{\mathbb{C}}^2$, we study the question whether the group generated by $f$ and the triangular subgroup is equal to the whole group $\mathrm{Aut}(\mathbb{A}_{\mathbb{C}}^2)$.

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arXiv2017

Counterexamples to the Complement Problem

We provide explicit counterexamples to the so-called Complement Problem in every dimension $n\geq3$, i.e. pairs of non-isomorphic irreducible hypersurfaces $H_1, H_2\subset\mathbb{C}^{n}$ whose complements $\mathbb{C}^{n}\setminus H_1$ and $\mathbb{C}^{n}\setminus H_2$ are isomorphic. Since we can arrange that one of the hypersurfaces is singular whereas the other is smooth, we also have counterexamples in the analytic setting.

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arXiv2016

The density property for Gizatullin surfaces of type $[[0,0,-r_2,-r_3]]$

Gizatullin surfaces of type $[[0,0,-r_2,-r_3]]$ can be described by the equations $yu = x P(x)$, $xv = u Q(u)$ and $yv = P(x) Q(u)$ in $\mathbb{C}^4_{x,y,u,v}$ where $P$ and $Q$ are non-constant polynomials. We establish the algebraic density property for smooth Gizatullin surfaces of this type. Moreover we also prove the density property for smooth surfaces given by these equations when $P$ and $Q$ are holomorphic functions.

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arXiv2015

Affine-ruled varieties without the Laurent cancellation property

We describe a method to construct hypersurfaces of the complex affine $n$-space with isomorphic $\mathbb{C}^*$-cylinders. Among these hypersurfaces, we find new explicit counterexamples to the Laurent Cancellation Problem, i.e. hypersurfaces that are non isomorphic, although their $\mathbb{C}^*$-cylinders are isomorphic as abstract algebraic varieties. We also provide examples of non isomorphic varieties $X$ and $Y$ with isomorphic cartesian squares $X\times X$ and $Y\times Y$.

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arXiv2014

Extension of automorphisms of rational smooth affine curves

We provide the existence, for every complex rational smooth affine curve $Γ$, of a linear action of $\mathrm{Aut}(Γ)$ on the affine 3-dimensional space $\mathbb{A}^3$, together with a $\mathrm{Aut}(Γ)$-equivariant closed embedding of $Γ$ into $\mathbb{A}^3$. It is not possible to decrease the dimension of the target, the reason for this obstruction is also precisely described.

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arXiv2014

On the closure of the tame automorphism group of affine three-space

We provide explicit families of tame automorphisms of the complex affine three-space which degenerate to wild automorphisms. This shows that the tame subgroup of the group of polynomial automorphisms of $\C^3$ is not closed, when the latter is seen as an infinite dimensional algebraic group.tomorphism group of affine three-space

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arXiv2013

Automorphism groups of certain rational hypersurfaces in complex four-space

The Russell cubic is a smooth contractible affine complex threefold which is not isomorphic to affine three-space. In previous articles, we discussed the structure of the automorphism group of this variety. Here we review some consequences of this structure and generalize some results to other hypersurfaces which arise as deformations of Koras-Russell threefolds.

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arXiv2013

A note on the stable equivalence problem

We provide counterexamples to the stable equivalence problem in every dimension $d\geq2$. That means that we construct hypersurfaces $H_1, H_2\subset\mathbb{C}^{d+1}$ whose cylinders $H_1\times\mathbb{C}$ and $H_2\times\mathbb{C}$ are equivalent hypersurfaces in $\mathbb{C}^{d+2}$, although $H_1$ and $H_2$ themselves are not equivalent by an automorphism of $\mathbb{C}^{d+1}$. We also give, for every $d\geq2$, examples of two non-isomorphic algebraic varieties of dimension $d$ which are biholomorphic.

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arXiv2010

Non cancellation for smooth contractible affine threefolds

We construct two non isomorphic contractible affine threefolds X and Y with isomorphic cylinders, showing that the generalized Cancellation Problem has a negative answer in general for contractible affine threefolds. We also establish that X and Y are actually biholomorphic as complex analytic varieties, providing the first example of a pair of biholomorphic but not isomorphic exotic affine 3-spaces.

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arXiv2009

Classification(s) of Danielewski hypersurfaces

The Danielewski hypersurfaces are the hypersurfaces $X_{Q,n}$ in $\mathbb{C}^3$ defined by an equation of the form $x^ny=Q(x,z)$ where $n\geq1$ and $Q(x,z)$ is a polynomial such that $Q(0,z)$ is of degree at least two. They were studied by many authors during the last twenty years. In the present article, we give their classification as algebraic varieties. We also give their classification up to automorphism of the ambient space. As a corollary, we obtain that every Danielewski hypersurface $X_{Q,n}$ with $n\geq2$ admits at least two non-equivalent embeddings into $\mathbb{C}^3$.

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arXiv2009

Inequivalent embeddings of the Koras-Russell cubic threefold

The Koras-Russell threefold is the hypersurface X of the complex affine four-space defined by the equation x^2y+z^2+t^3+x=0. It is well-known that X is smooth contractible and rational but that it is not algebraically isomorphic to affine three-space. The main result of this article is to show that there exists another hypersurface Y of the affine four-space, which is isomorphic to X as an abstract variety, but such that there exists no algebraic automorphism of the ambient space which restricts to an isomorphism between X and Y. In other words, the two hypersurfaces are inequivalent. The proof of this result is based on the description of the automorphism group of X. We show in particular that all algebraic automorphisms of X extend to automorphisms of the ambient space.

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arXiv2008

The Nagata automorphism is shifted linearizable

A polynomial automorphism $F$ is called {\em shifted linearizable} if there exists a linear map $L$ such that $LF$ is linearizable. We prove that the Nagata automorphism $N:=(X-YΔ-ZΔ^2,Y+ZΔ, Z)$ where $Δ=XZ+Y^2$ is shifted linearizable. More precisely, defining $L_{(a,b,c)}$ as the diagonal linear map having $a,b,c$ on its diagonal, we prove that if $ac=b^2$, then $L_{(a,b,c)}N$ is linearizable if and only if $bc\not = 1$. We do this as part of a significantly larger theory: for example, any exponent of a homogeneous locally finite derivation is shifted linearizable. We pose the conjecture that the group generated by the linearizable automorphisms may generate the group of automorphisms, and explain why this is a natural question.

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arXiv2006

On a class of Danielewski surfaces in affine 3-space

L. Makar-Limanov computed the automorphisms groups of surfaces in $\mathbb{C}^{3}$ defined by the equations $x^{n}z-P(y)=0$, where $n\geq1$ and $P(y)$ is a nonzero polynomial. Similar results have been obtained by A. Crachiola for surfaces defined by the equations $x^{n}z-y^{2}-h(x)y=0$, where $n\geq2$ and $h(0)\neq0$, defined over an arbitrary base field. Here we consider the more general surfaces defined by the equations $x^{n}z-Q(x,y)=0$, where $n\geq2$ and $Q(x,y)$ is a polynomial with coefficients in an arbitrary base field $k$. Among these surfaces, we characterize the ones which are Danielewski surfaces and we compute their automorphism groups. We study closed embeddings of these surfaces in affine 3-space. We show that in general their automorphisms do not extend to the ambient space. Finally, we give explicit examples of $\mathbb{C}^{*}$-actions on a surface in $\mathbb{C}^{3}$ which can be extended holomorphically but not algebraically to a $\mathbb{C}^{*}$-action on $\mathbb{C}^{3}$.

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