A variety is said to satisfy Condition (A) if every finite abelian subgroup of its automorphism group has a fixed point. We show that a smooth Fano 3-fold not satisfying Condition (A) is K-polystable unless it is contained in eight exceptional deformation families (seven of them consist of one smooth member, and one of them has two-parameter moduli).

In this paper, we study finite subgroups G⊂Aut(ℙ^n) such that ℙ^n is G-birationally rigid. For each n⩾ 3, we prove that Aut(ℙ^n) contains at most finitely many such subgroups up to conjugation. For n=4, we prove that ℙ^4 is G-birationally superrigid if G≃PSp_4(𝐅_3).

A smooth variety is said to satisfy Condition (A) if every finite abelian subgroup of its automorphism group has a fixed point. We classify smooth Fano 3-folds that satisfy Condition (A).

We study the connectedness of the real locus of smooth geometrically rational Fano threefolds and prove a sufficient criterion of ℝ-rationality.

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 ℙ^5 along a conic. We also describe the automorphism groups of these threefolds.

This review is an elaboration of a presentation given at the Real algebraic geometry and singularities conference in honor of Wojciech Kucharz's 70th birthday in Krakow in 2022.

We explore connections between existence of -rational points for Fano varieties defined over , a subfield of ℂ, and existence of Kähler-Einstein metrics on their geometric models. First, we show that geometric models of del Pezzo surfaces with at worst quotient singularities defined over ⊂ℂ admit (orbifold) Kähler–Einstein metrics if they do not have -rational points. Then we prove the same result for smooth Fano 3-folds with 8 exceptions. Consequently, we explicitly describe several families of pointless Fano 3-folds whose geometric models admit Kähler-Einstein metrics. In particular, we obtain new examples of prime Fano 3-folds of genus 12 that admit Kähler–Einstein metrics. Our result can also be used to prove existence of rational points for certain Fano varieties, for example for any smooth Fano 3-fold over ⊂ℂ whose geometric model is strictly K-semistable.

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.

This volume originated as a proceedings of the conference on Algebraic Geometry organized by four of us at the University of Auckland (Auckland, New Zealand) back in December 2019. This was the first conference on Algebraic Geometry that was ever organized in New Zealand. The New Zealand conference was a great success. Thirty four mathematicians participated in it:

We give a sufficient condition in order that n closed connected subsets in the n-dimensional real projective space admit a common multitangent hyperplane.

Most of this chapter was previously published in [Man14] in the “Gazette de la SMF”.

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 present the topological classification of real parts of real regular elliptic surfaces with a real section. MSC 2000: 14P25 14J27

We study real rational models of the euclidean affine 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, $\mathbb{P}^{2}(\mathbb{R})$ is the only model. A fake real plane is a smooth geometrically integral surface $S$ defined over $\mathbb{R}$ not isomorphic to $\mathbb{A}^2_\mathbb{R}$, whose real locus $S(\mathbb{R})$ is diffeomorphic to $\mathbb{R}^2$ and such that the complex surface $S_\mathbb{C}(\mathbb{C})$ has the rational homology type of $\mathbb{A}^2_\mathbb{C}$. We prove that fake planes exist by giving many examples and we tackle the question: does there exist fake planes $S$ such that $S(\mathbb{R})$ is not birationally diffeomorphic to $\mathbb{A}^2_\mathbb{R}(\mathbb{R})$?

We survey some results on real rational surfaces focused on their topology and their birational geometry.

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 Nullstellen-satz. 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.

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 exhibit examples of such planes of every Kodaira dimension \(\kappa \in \{-\infty ,0,1,2\}\) which are birationally diffeomorphic to \(\mathbb {R}^{2}\).