arXiv subjects
Publications and source records attributed to Jérémy Blanc.
We give a presentation by generators and relations of the group of birational transformations of the plane preserving a differential form with poles along a triple line. This group strictly contains the automorphism group of the affine plane.
We prove that any group of cardinality at most the one of $\mathbb{C}$ is a quotient of any Cremona group of rank at least $4$. This provides a definitive answer to the question of what the quotients of Cremona groups can be. As a consequence, this gives a negative answer to the question of I. Dolgachev of whether Cremona groups of all ranks are generated by involutions. As another application, we show that higher Cremona groups do not enjoy some classical group-theoretic properties (namely, the Hopfian property) which are satisfied by Cremona groups of rank $2$. Finally, we discover that the $3$-torsion of the Cremona group of rank at least $4$ is not countable. To deduce these properties of higher Cremona groups, we first describe the group of birational transformations of a non-trivial Severi-Brauer surface $S$ over a perfect field, proving in particular that if $S$ contains a point of degree $6$, then its group of birational self-maps is not generated by elements of finite order as it admits a surjective group homomorphism to~$\mathbb{Z}$. We then use this result to study Mori fibre spaces over the field of complex numbers, for which the generic fibre is a non-trivial Severi-Brauer surface.
In this article we give two explicit families of automorphisms of degree $\leq 3$ of the affine $3$-space $\mathbb{A}^3$ such that each automorphism of degree $\leq 3$ of $\mathbb{A}^3$ is a member of one of these families up to composition of affine automorphisms at the source and target; this shows in particular that all of them are tame. As an application, we give the list of all dynamical degrees of automorphisms of degree $\leq 3$ of $\mathbb{A}^3$; this is a set of $3$ integers and $9$ quadratic integers. Moreover, we also describe up to compositions with affine automorphisms for $n\geq 1$ all morphisms $\mathbb{A}^3 \to \mathbb{A}^n$ of degree $\leq 3$ with the property that the preimage of every affine hyperplane in $\mathbb{A}^n$ is isomorphic to $\mathbb{A}^2$.
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.
In this paper we provide the complete classification of $\mathbb{P}^1$-bundles over smooth projective rational surfaces whose neutral component of the automorphism group is maximal. Our results hold over any algebraically closed field of characteristic zero.
We study the connected algebraic groups acting on Mori fibrations $X \to Y$ with $X$ a rational threefold and $\mathrm{dim}(Y) \geq 1$. More precisely, for these fibre spaces we consider the neutral component of their automorphism groups and study their equivariant birational geometry. This is done using, inter alia, minimal model program and Sarkisov program and allows us to determine the maximal connected algebraic subgroups of $\mathrm{Bir}(\mathbb{P}^3)$, recovering most of the classification results of Hiroshi Umemura in the complex case.
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.
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 Jonquières elements.
Let $A$ be an abelian variety over an algebraically closed field. We show that $A$ is the automorphism group scheme of some smooth projective variety if and only if $A$ has only finitely many automorphisms as an algebraic group. This generalizes a result of Lombardo and Maffei for complex abelian varieties.
We study the possible dynamical degrees of automorphisms of the affine space $\mathbb{A}^n$. In dimension $n=3$, we determine all dynamical degrees arising from the composition of an affine automorphism with a triangular one. This generalises the easier case of shift-like automorphisms which can be studied in any dimension. We also prove that each weak Perron number is the dynamical degree of an affine-triangular automorphism of the affine space $\mathbb{A}^n$ for some $n$, and we give the best possible $n$ for quadratic integers, which is either $3$ or $4$.
We study the complexity of birational self-maps of a projective threefold $X$ by looking at the birational type of surfaces contracted. These surfaces are birational to the product of the projective line with a smooth projective curve. We prove that the genus of the curves occuring is unbounded if and only if $X$ is birational to a conic bundle or a fibration into cubic surfaces. Similarly, we prove that the gonality of the curves is unbounded if and only if $X$ is birational to a conic bundle.
Let $X/\mathbb{P}^1$ be a Mori fibre space with general fibre of Picard rank at least two. We prove that there is a proper closed subset $S\subsetneq X$, invariant by the connected component of the identity ${\rm Aut}^{\circ}(X)$ of the automorphism group of $X$, which is moreover the orbit of a section $s$ and whose intersection with a fibre is an orbit of the subgroup of ${\rm Aut}^{\circ}(X)$ acting trivially on $\mathbb{P}^1$. Such result is a tool to describe equivariant birational maps from $X/\mathbb{P}^1$ to other Mori fibre spaces and therefore finds its applications in the study of connected algebraic subgroups of ${\rm Aut}^{\circ}(X)$. This represents a first reduction step towards a possible classification of maximal connected algebraic subgroups of the Cremona group of rank $4$.
We prove that the group of birational transformations of a Del Pezzo fibration of degree 3 over a curve is not simple, by giving a surjective group homomorphism to a free product of infinitely many groups of order 2. As a consequence we also obtain that the Cremona group of rank 3 is not generated by birational maps preserving a rational fibration. Besides, its subgroup generated by all connected algebraic subgroups is a proper normal subgroup.
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 $X_f$.
This article describes the geometry of isomorphisms between complements of geometrically irreducible closed curves in the affine plane $\mathbb{A}^2$, over an arbitrary field, which do not extend to an automorphism of $\mathbb{A}^2$. We show that such isomorphisms are quite exceptional. In particular, they occur only when both curves are isomorphic to open subsets of the affine line $\mathbb{A}^1$, with the same number of complement points, over any field extension of the ground field. Moreover, the isomorphism is uniquely determined by one of the curves, up to left composition with an automorphism of $\mathbb{A}^2$, except in the case where the curve is isomorphic to the affine line $\mathbb{A}^1$ or to the punctured line $\mathbb{A}^1 \setminus \{0\}$. If one curve is isomorphic to $\mathbb{A}^1$, then both curves are in fact equivalent to lines. In addition, for any positive integer $n$, we construct a sequence of $n$ pairwise non-equivalent closed embeddings of $\mathbb{A}^1 \setminus \{0\}$ with isomorphic complements. In characteristic~$0$ we even construct infinite sequences with this property. Finally, we give a geometric construction that produces a large family of examples of non-isomorphic geometrically irreducible closed curves in $\mathbb{A}^2$ that have isomorphic complements, answering negatively the Complement Problem posed by Hanspeter Kraft.. This also gives a negative answer to the holomorphic version of this problem in any dimension $n \geq 2$. The question had been raised by Pierre-Marie Poloni.
The Cremona group is the group of birational transformations of the plane. A birational transformation for which there exists a pencil of lines which is sent onto another pencil of lines is called a Jonquières transformation. By the famous Noether-Castelnuovo theorem, every birational transformation $f$ is a product of Jonquières transformations. The minimal number of factors in such a product will be called the length, and written $\mathrm{lgth}(f)$. Even if this length is rather unpredictable, we provide an explicit algorithm to compute it, which only depends on the multiplicities of the linear system of $f$. As an application of this computation, we give a few properties of the dynamical length of $f$ defined as the limit of the sequence $n \mapsto \mathrm{lgth} (f^n) / n$. It follows for example that an element of the Cremona group is distorted if and only if it is algebraic. The computation of the length may also be applied to the so called Wright complex associated with the Cremona group: This has been done recently by Lonjou. Moreover, we show that the restriction of the length to the automorphism group of the affine plane is the classical length of this latter group (the length coming from its amalgamated structure). In another direction, we compute the lengths and dynamical lengths of all monomial transformations, and of some Halphen transformations. Finally, we show that the length is a lower semicontinuous map on the Cremona group endowed with its Zariski topology.
We introduce a new invariant, the real (logarithmic)-Kodaira dimension, that allows to distinguish smooth real algebraic surfaces up to birational diffeomorphism. As an application, we construct infinite families of smooth rational real algebraic surfaces with trivial homology groups, whose real loci are diffeomorphic to $\mathbb{R}^2$, but which are pairwise not birationally diffeomorphic. There are thus infinitely many non-trivial models of the euclidean plane, contrary to the compact case.
We show that the only finite quasi-simple non-abelian groups that can faithfully act on rationally connected threefolds are the following groups: $\mathfrak{A}_5$, $\operatorname{PSL}_2(\mathbf{F}_7)$, $\mathfrak{A}_6$, $\operatorname{SL}_2(\mathbf{F}_8)$, $\mathfrak{A}_7$, $\operatorname{PSp}_4(\mathbf{F}_3)$, $\operatorname{SL}_2(\mathbf{F}_{7})$, $2.\mathfrak{A}_5$, $2.\mathfrak{A}_6$, $3.\mathfrak{A}_6$ or $6.\mathfrak{A}_6$. All of these groups with a possible exception of $2.\mathfrak{A}_6$ and $6.\mathfrak{A}_6$ indeed act on some rationally connected threefolds.