Rational Varieties Research Articles

Unramified Brauer group of quotient spaces by finite groups

We give a general procedure to determine the unramified Brauer group of quotients of rational varieties by finite groups.

Complete intersection hyperkähler fourfolds with respect to equivariant vector bundles over rational homogeneous varieties of Picard number one

Complete intersection hyperkähler fourfolds with respect to equivariant vector bundles over rational homogeneous varieties of Picard number one

Reciprocity obstruction to strong approximation over p-adic function fields

Over function fields of p-adic curves, we construct stably rational varieties in the form of homogeneous spaces of SLn with semisimple simply connected stabilizers and we show that strong approximation away from a non-empty set of places fails for such varieties. The construction combines the Lichtenbaum duality and the degree 3 cohomological invariants of the stabilizers. We then establish a reciprocity obstruction which accounts for this failure of strong approximation. We show that this reciprocity obstruction to strong approximation is the only one for counterexamples we constructed, and also for classifying varieties of tori. We also show that this reciprocity obstruction to strong approximation is compatible with known results for tori. At the end, we explain how a similar point of view shows that the reciprocity obstruction to weak approximation is the only one for classifying varieties of tori over p-adic function fields.

On Massey products and rational homogeneous varieties

On Massey products and rational homogeneous varieties

Varieties covered by affine spaces, uniformly rational varieties and their cones

It was shown in Kaliman snd Zaidenberg (2023) [26] that the affine cones over flag manifolds and rational smooth projective surfaces are elliptic in the sense of Gromov. The latter remains true after successive blowups of points on these varieties. In the present article we extend this to smooth projective spherical varieties (in particular, toric varieties) successively blown up along smooth subvarieties. The same holds, more generally, for uniformly rational projective varieties, in particular, for projective varieties covered by affine spaces. It occurs also that stably uniformly rational complete varieties are elliptic.

Inversion maps and torus actions on rational homogeneous varieties

Complex projective algebraic varieties with {{mathbb {C}}}^*-actions can be thought of as geometric counterparts of birational transformations. In this paper we describe geometrically the birational transformations associated to rational homogeneous varieties endowed with a {{mathbb {C}}}^*-action with no proper isotropy subgroups.

Approximation and homotopy in regulous geometry

Let $X$ , $Y$ be nonsingular real algebraic sets. A map $\varphi \colon X \to Y$ is said to be $k$ -regulous, where $k$ is a nonnegative integer, if it is of class $\mathcal {C}^k$ and the restriction of $\varphi$ to some Zariski open dense subset of $X$ is a regular map. Assuming that $Y$ is uniformly rational, and $k \geq 1$ , we prove that a $\mathcal {C}^{\infty }$ map $f \colon X \to Y$ can be approximated by $k$ -regulous maps in the $\mathcal {C}^k$ topology if and only if $f$ is homotopic to a $k$ -regulous map. The class of uniformly rational real algebraic varieties includes spheres, Grassmannians and rational nonsingular surfaces, and is stable under blowing up nonsingular centers. Furthermore, taking $Y=\mathbb {S}^p$ (the unit $p$ -dimensional sphere), we obtain several new results on approximation of $\mathcal {C}^{\infty }$ maps from $X$ into $\mathbb {S}^p$ by $k$ -regulous maps in the $\mathcal {C}^k$ topology, for $k \geq 0$ .

Effective good divisibility of rational homogeneous varieties

Effective good divisibility of rational homogeneous varieties

Maximal disjoint Schubert cycles in rational homogeneous varieties

AbstractIn this paper, we study properties of the Chow ring of rational homogeneous varieties of classical type, more concretely, effective zero divisors of low codimension, and a related invariant called effective good divisibility. This information is then used to study the question of (non)existence of nonconstant maps among these varieties, generalizing previous results for projective spaces and Grassmannians.

Optical soliton propagation in the Benjamin–Bona–Mahoney–Peregrine equation using two analytical schemes

The purpose of this research is to investigate the nonlinear partial differential equation known as Benjamin–Bona–Mahoney–Peregrine equation. Two versatile techniques, namely the extended (G′G2)-expansion method and modified auxiliary equation method is used to extract the exact solutions of the nonlinear Benjamin–Bona–Mahoney–Peregrine equation. Consequently, a diverse range of novel collections of exact traveling wave solutions are identified such as kink, bright, singular kink and periodic soliton solution. To provide a physical understanding, several of the discovered solutions are shown in figures. The resulting solutions are of the trigonometric, hyperbolic, and rational varieties and the stability analysis are also discussed. In addition to this, the graphical aspects of the produced solutions are investigated. Due to the vital necessity of this aspect, we make use of the suitable parameter values to place a priority on the physical aspects of the provided findings by utilizing 3D and contour charts. The suggested procedures are of great use and can be extended to additional nonlinear evolutionary equations that reflect nonlinear physical frameworks in the nonlinear sciences.

Kähler-Ricci flow on rational homogeneous varieties

In this work, we study the Kähler-Ricci flow on rational homogeneous varieties exploring the interplay between projective algebraic geometry and representation theory which underlies the classical Borel-Weil theorem. By using elements of representation theory of semisimple Lie groups and Lie algebras, we give an explicit description for all solutions of the Kähler-Ricci flow with homogeneous initial condition. This description enables us to compute explicitly the maximal existence time for any solution starting at a homogeneous Kähler metric and obtain explicit upper and lower bounds for several geometric quantities along the flow, including curvatures, volume, diameter, and the first non-zero eigenvalue of the Laplacian. As an application of our main result, we investigate the relationship between certain numerical invariants associated with ample divisors and numerical invariants arising from solutions of the Kähler-Ricci flow in the homogeneous setting.

Embeddings of Automorphism Groups of Free Groups into Automorphism Groups of Affine Algebraic Varieties

For every integer n>0, we construct a new infinite series of rational affine algebraic varieties such that their automorphism groups contain the automorphism group mathrm{Aut}(F_n) of the free group F_n of rank n and the braid group B_n on n strands. The automorphism groups of such varieties are nonlinear for ngeq 3 and are nonamenable for ngeq 2. As an application, we prove that every Cremona group of rank {geq},3n-1 contains the groups mathrm{Aut}(F_n) and B_n. This bound is 1 better than the bound published earlier by the author; with respect to B_n, the order of its growth rate is one less than that of the bound following from a paper by D. Krammer. The construction is based on triples (G,R,n), where G is a connected semisimple algebraic group and R is a closed subgroup of its maximal torus.

Deformations of Hyperelliptic and Generalized Hyperelliptic Polarized Varieties

The purpose of this article is twofold. Firstly, we address and completely solve the following question: Let (X, L) be a smooth, hyperelliptic polarized variety and let varphi : X longrightarrow Y subset textbf{P}^N be the morphism induced by |L|; when does varphi deform to a birational map? Secondly, we introduce the notion of “generalized hyperelliptic varieties” and carry out a study of their deformations. Regarding the first topic, we settle the non trivial, open cases of (X, L) being Fano-K3 and of (X, L) having dimension m ge 2, sectional genus g and L^m=2g. This was not addressed by Fujita in his study of hyperelliptic polarized varieties and requires the introduction of new methods and techniques to handle it. In the Fano-K3 case, all deformations of (X, L) are again hyperelliptic except if Y is a hyperquadric. By contrast, in the L^m=2g case, with one exception, a general deformation of varphi is a finite birational morphism. This is especially interesting and unexpected because, in the light of earlier results, varphi rarely deforms to a birational morphism when Y is a rational variety, as is our case. The Fano-K3 case contrasts with canonical morphisms of hyperelliptic curves and with hyperelliptic K3 surfaces of genus g ge 3. Regarding the second topic, we completely answer the question for generalized hyperelliptic polarized Fano and Calabi–Yau varieties. For generalized hyperelliptic varieties of general type we do this in even greater generality, since our result holds for Y toric. Standard methods in deformation theory do not work in the present setting. Thus, to settle these long standing open questions, we bring in new ideas and techniques building on those introduced by the authors concerning deformations of finite morphisms and the existence and smoothings of certain multiple structures. We also prove a new general result on unobstructedness of morphisms that factor through a double cover and apply it to the case of generalized hyperelliptic varieties.

On a family of linear MRD codes with parameters $$[8\times 8,16,7]_q$$

In this paper we consider a family \(\mathcal {F}\) of 2n-dimensional \({\mathbb {F}}_q\)-linear rank metric codes in \({\mathbb {F}}_q^{n\times n}\) arising from polynomials of the form \(x^{q^s}+\delta x^{q^{\frac{n}{2}+s}}\in {\mathbb {F}}_{q^n}[x]\). The family \(\mathcal {F}\) was introduced by Csajbók et al. (JAMA 548:203–220) as a potential source for maximum rank distance (MRD) codes. Indeed, they showed that \(\mathcal {F}\) contains MRD codes for \(n=8\), and other subsequent partial results have been provided in the literature towards the classification of MRD codes in \(\mathcal {F}\) for any n. In particular, the classification has been reached when n is smaller than 8, and also for n greater than 8 provided that s is small enough with respect to n. In this paper we deal with the open case \(n=8\), providing a classification for any large enough odd prime power q. The techniques are from algebraic geometry over finite fields, since our strategy requires the analysis of certain 3-dimensional \({\mathbb {F}}_q\)-rational algebraic varieties in a 7-dimensional projective space. We also show that the MRD codes in \(\mathcal {F}\) are not equivalent to any other MRD codes known so far.

Сучасний стан та організація харчування військовослужбовців Збройних Сил України в умовах російської агресії проти України

Background. In order to ensure a high level of physical, cognitive, and emotional state of servicemen, to perform service and combat tasks, it is necessary to provide adequate nutrition for the personnel, to establish the logistics of the supply of food products in field conditions away from the supply bases. Aim: to analyze the requirements for the organization of nutrition for servicemen of the Armed Forces of Ukraine, including during the execution of combat missions. Materials and methods. information retrieval, content analysis, analysis of legislative and regulatory documents and scientific publications of ukrainian and foreign researchers. Results: Since the first days of the full-scale invasion of Ukraine by Russian troops, a mobile system of troop nutrition has been developed, in which an important place is occupied by the implementation of a new approach to food rationing based on the creation of rations that can provide military personnel with food that is as prepared as possible for consumption. The transition to the new food system in accordance with the Catalog allows us to organize high-quality nutrition for personnel using a variety of food groups and the possibility of individual adjustment to the conditions of stay and the tasks of the military unit. In the absence of cooking conditions, the military eat dry rations: general military rations (Norm № 10) or more advanced and balanced rations (Norm № 15). The variety of rations, their equipment, technological (retort packs) and tactical (flameless heaters) innovations are a significant step towards optimizing the nutrition of the military personnel of the Armed Forces of Ukraine. Conclusions: Despite numerous innovations in the system of catering for military personnel, including during combat operations, constant nutritional monitoring is needed to adjust military rations to ensure that personnel of various specialties are provided with adequate nutrition.

Imagining the Ring of Gyges. The Dual Rationality of Thought-Experimenting

In her already classical criticism of thought-experimenting, Kathy Wilkes points to superficialities in the most famous moral-political thought experiments, taking the Ring of Gyges as her central example. Her critics defend the Ring by discussing possible variations in the scenario(s) imagined. I propose here that the debate points to a significant dual structure of thought experiments. Their initial presentation(s) mobilize the immediate, cognitively not very impressive imaginative and refl ective efforts both of the proponent and the listener of the proposal. The further debate, like the one exemplifi ed by Wilkes’s criticisms and some of the answers, appeals to a deeper, more rational variety of imagination and reasoning. I suggest that this duality is typical for moral and political thought experimenting in general, conjecture that it might be extended to the whole area of thought experimenting.

Curve valuations and mixed volumes in the implicitization of rational varieties

We address the description of the tropicalization of families of rational varieties under parametrizations with prescribed support, via curve valuations. We recover and extend results by Sturmfels, Tevelev and Yu for generic coefficients, considering rational parametrizations with non-trivial denominator. The advantage of our point of view is that it can be generalized to deal with non-generic parametrizations. We provide a detailed analysis of the degree of the closed image, based on combinatorial conditions on the relative positions of the supports of the polynomials defining the parametrization. We obtain a new formula and finer bounds on the degree, when the supports of the polynomials are different. We also present a new formula and bounds for the order at the origin in case the closed image is a hypersurface.

Amplified endomorphisms of Fano fourfolds

Let X be a smooth Fano fourfold admitting a conic bundle structure. We show that X is toric if and only if X admits an amplified endomorphism; in this case, X is a rational variety.

RATIONALITY OVER NONCLOSED FIELDS OF FANO THREEFOLDS WITH HIGHER GEOMETRIC PICARD RANK

AbstractWe prove rationality criteria over nonclosed fields of characteristic $0$ for five out of six types of geometrically rational Fano threefolds of Picard number $1$ and geometric Picard number bigger than $1$ . For the last type of such threefolds, we provide a unirationality criterion and construct examples of unirational but not stably rational varieties of this type.

On the existence of finite surjective parametrizations of affine surfaces

We investigate surjective parametrizations of rational algebraic varieties, in the vein of recent work by Jorge Caravantes, J. Rafael Sendra, David Sevilla, and Carlos Villarino. In particular, we show how to construct plenty of examples of affine surfaces S not admitting a finite surjective morphism f:A2→S.