arXiv subjects
Publications and source records attributed to Frédéric Mangolte.
We study K-stability of smooth Fano threefolds of Picard rank $2$ and degree $22$ which can be obtained by blowing up a smooth complete intersection of two quadrics in $\mathbb{P}^5$ along a conic. We also describe the automorphism groups of these threefolds.
We study the connectedness of the real locus of smooth geometrically rational Fano threefolds and prove a sufficient criterion of $\mathbb{R}$-rationality.
This text is contribution 77 to the ZAG Handbook of Modern Algebraic Geometry, edited by I. Cheltsov and J. Martinez-Garcia, and summarises the Short Communication I presented at the Geometry and Topology Session of the International Congress of Mathematicians which took place at the University of Copenhagen in 2022.
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 4 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.
We give a sufficient condition in order that $n$ closed connected subsets in the $n$-dimensional real projective space admit a common multitangent hyperplane.
In Dubouloz and Mangolte (Fake real planes: exotic affine algebraic models of $\mathbb{R}^{2}$, arXiv:1507.01574, 2015) we define and partially classify fake real planes, that is, minimal complex surfaces with conjugation whose real locus is diffeomorphic to the euclidean real plane $\mathbb{R}^{2}$. Classification results are given up to biregular isomorphisms and up to birational diffeomorphisms. In this note, we describe in an elementary way numerous examples of fake real planes and we exhibit examples of such planes of every Kodaira dimension $κ\in \{-\infty,0,1,2\}$ which are birationally diffeomorphic to $\mathbb{R}^{2}$.
We study real rational models of the euclidean plane $\mathbb{R}^2$ up to isomorphisms and up to birational diffeomorphisms. The analogous study in the compact case, that is the classification of real rational models of the real projective plane $\mathbb{R}\mathbb{P}^2$ is well known: up to birational diffeomorphisms, there is only one model. A fake real plane is a nonsingular affine surface defined over the reals with homologically trivial complex locus and real locus diffeomorphic to $\mathbb{R}^2$ but which is not isomorphic to the real affine plane. We prove that fake planes exist by giving many examples and we tackle the question: does there exist fake planes whose real locus is not birationally diffeomorphic to the real affine plane?
We study smooth rational closed embeddings of the real affine line into the real affine plane, that is algebraic rational maps from the real affine line to the real affine plane which induce smooth closed embeddings of the real euclidean line into the real euclidean plane. We consider these up to equivalence under the group of birational automorphisms of the real affine plane which are diffeomorphisms of its real locus. We show that in contrast with the situation in the categories of smooth manifolds with smooth maps and of real algebraic varieties with regular maps where there is only one equivalence class up to isomorphism, there are non-equivalent smooth rational closed embeddings up to such birational diffeomorphisms.
We address the problem of the maximal finite number of real points of a real algebraic curve (of a given degree and, sometimes, genus) in the projective plane. We improve the known upper and lower bounds and construct close to optimal curves of small degree. Our upper bound is sharp if the genus is small as compared to the degree. Some of the results are extended to other real algebraic surfaces, most notably ruled.
We study the ring of rational functions admitting a continuous extension to the real affine space. We establish several properties of this ring. In particular, we prove a strong Nullstelensatz. We study the scheme theoretic properties and prove regulous versions of Theorems A and B of Cartan. We also give a geometrical characterization of prime ideals of this ring in terms of their zero-locus and relate them to euclidean closed Zariski-constructible sets.
We give necessary and sufficient topological conditions for a simple closed curve on a real rational surface to be approximable by smooth rational curves. We also study approximation by smooth rational curves with given complex self-intersection number.
We give an explicit set of generators for various natural subgroups of the real Cremona group Bir_R(P^2). This completes and unifies former results by several authors.
We review the state of the art of the classification of real uniruled threefolds
A group G acts infinitely transitively on a set Y if for every positive integer m, its action is m-transitive on Y. Given a real affine algebraic variety Y of dimension greater than or equal to two, we show that, under a mild restriction, if the special automorphism group of Y (the group generated by one-parameter unipotent subgroups) is infinitely transitive on each connected component of the smooth locus of Y, then for any real affine suspension X over Y, the special automorphism group of X is infinitely transitive on each connected component of the smooth locus of X. This generalizes a recent result by Arzhantsev, Kuyumzhiyan and Zaidenberg over the field of real numbers.
In this article we study the transitivity of the group of automorphisms of real algebraic surfaces. We characterize real algebraic surfaces with very transitive automorphism groups. We give applications to the classification of real algebraic models of compact surfaces: these applications yield new insight into the geometry of the real locus, proving several surprising facts on this geometry. This geometry can be thought of as a half-way point between the biregular and birational geometries.
We prove using symplectic field theory that if the suspension of a hyperbolic diffeomorphism of the two-torus Lagrangian embeds in a closed uniruled symplectic six-manifold, then its image contains the boundary of a symplectic disc with vanishing Maslov index. This prevents such a Lagrangian submanifold to be monotone, for instance the real locus of a smooth real Fano manifold. It also prevents any Sol manifold to be in the real locus of an orientable real Del Pezzo fibration over a curve, confirming an expectation of J. Kollár. Finally, it constraints Hamiltonian diffeomorphisms of uniruled symplectic four-manifolds.
We show that the action of Cremona transformations on the real points of quadrics exhibits the full complexity of the diffeomorphisms of the sphere, the torus, and of all non-orientable surfaces. The main result says that if X is rational, then Aut(X), the group of algebraic automorphisms, is dense in Diff(X), the group of self-diffeomorphisms of X.
Let X be a singular real rational surface obtained from a smooth real rational surface by performing weighted blow-ups. Denote by Aut(X) the group of algebraic automorphisms of X into itself. Let n be a natural integer and let e=[e_1,...,e_l] be a partition of n. Denote by X^e the set of l-tuples (P_1,...,P_l) of distinct nonsingular curvilinear infinitely near points of X of orders (e_1,...,e_l). We show that the group Aut(X) acts transitively on X^e. This statement generalizes earlier work where the case of the trivial partition e=[1,...,1] was treated under the supplementary condition that X is nonsingular. As an application we classify singular real rational surfaces obtained from nonsingular surfaces by performing weighted blow-ups.