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Affine cones over smooth cubic surfaces
Journal of the European Mathematical Society. 2016. Vol. 18. No. 7. P. 1537–1564.
Cheltsov I., Park J., Won J.
We show that affine cones over smooth cubic surfaces do not admit non-trivial Ga-actions.
Perepechko A., Taiwanese Journal of Mathematics 2025 Vol. 29 No. 6 P. 1633–1650
Generic flexibility of affine cones over Fano varieties is a subject of active study recently. For del Pezzo surfaces the question is completely studied in degree at least 3, and partially in degree 2.
We present a Sagemath module that facilitates most operations for verifying the generic flexibility of affine cones over del Pezzo surfaces and ...
Added: February 16, 2026
Arzhantsev I., Proceedings of the Steklov Institute of Mathematics 2025 Vol. 329 P. 26–32
We prove that an affine cone X admits a surjective morphism from an affine space if and only if X is unirational. ...
Added: September 6, 2025
Ornea L., Verbitsky M., Journal of Geometry and Physics 2024 Vol. 198 Article 105103
A complex manifold X is called "LCK manifolds with potential" if it can be realized as a complex submanifold of a Hopf manifold. Let Y its $\Z$-covering, considered as a complex submanifold in Cn∖0. We prove that Y is algebraic. We call the manifolds obtained this way the algebraic cones, and show that the affine algebraic structure on Y is independent from the choice of X. ...
Added: December 2, 2024
Arzhantsev I., Труды Математического института им. В.А. Стеклова РАН 2025 Т. 329 С. 33–39
We prove that an affine cone X admits a surjective morphism from an affine space if and only if X is unirational. ...
Added: September 18, 2024
I. Arzhantsev, Kaliman S., M. Zaidenberg, Advances in Mathematics 2024 Vol. 437 Article 109449
It was shown by Kaliman and Zaidenberg (2023) 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 ...
Added: December 17, 2023
I. R. Lavrukhin, A. A. Yelizarov, S. V. Bashkevich, , in: 2023 Systems of Signal Synchronization, Generating and Processing in Telecommunications (SYNCHROINFO).: IEEE, 2023. P. 1–4.
This paper presents the results of a study of the radiation characteristics of antennas located on a cylindrical surface. The characteristics of the antennas including the directional pattern as well as the optimal amplitude distribution on the cylinder in parallel and perpendicular polarizations are determined. In addition, the paper focuses on the analysis of the ...
Added: October 20, 2023
Arzhantsev I., St Petersburg Mathematical Journal 2023 Vol. 34 No. 2 P. 143–178
We survey recent results on multiple transitivity for automorphism groups of affine algebraic varieties. We consider the property of infinite transitivity of the special automorphism group, which is equivalent to flexibility of the corresponding affine variety. These properties have important algebraic and geometric consequences. At the same time they are fulfilled for wide classes of ...
Added: March 30, 2023
Arzhantsev I., Алгебра и анализ 2022 Т. 34 № 2 С. 1–55
В работе дан обзор результатов последних лет о кратной транзитивности действий групп автоморфизмов аффинных алгебраических многообразий. Рассматривается свойство бесконечной транзитивности действия группы специальных автоморфизмов и эквивалентное ему свойство гибкости многообразия. Данные свойства имеют важные алгебраические и геометрические следствия, и при этом они выполнены для широких классов многообразий. Отдельно изучаются случаи, когда бесконечная транзитивность имеет место ...
Added: March 14, 2022
Cheltsov I., Park J., Prokhorov Y. et al., EMS Surveys in Mathematical Sciences 2021 Vol. 8 No. 1-2 P. 39–105
This paper is a survey about cylinders in Fano varieties and related problems ...
Added: November 16, 2021
Cheltsov I., Prokhorov Y., Algebraic Geometry 2021 Vol. 8 No. 3 P. 319–357
We classify del Pezzo surfaces with Du Val singularities that have infinite automorphism groups, and describe the connected components of their automorphisms groups. ...
Added: September 7, 2021
Cheltsov I., Martinez-Garcia J., Transactions of the American Mathematical Society 2019 Vol. 372 No. 10 P. 7255–7296
We provide new examples of -unstable polarized smooth del Pezzo surfaces using a flopped version first used by Cheltsov and Rubinstein of the test configurations introduced by Ross and Thomas. As an application, we provide new obstructions for the existence of constant scalar curvature Kähler metrics on polarized smooth del Pezzo surfaces. ...
Added: May 10, 2020
Cheltsov I., Kuznetsov A., Shramov K., Algebra & Number Theory 2020 Vol. 14 No. 1 P. 213–274
We construct two small resolutions of singularities of the Coble fourfold (the double cover of the four-dimensional projective space branched over the Igusa quartic). We use them to show that all 𝔖6-invariant three-dimensional quartics are birational to conic bundles over the quintic del Pezzo surface with the discriminant curves from the Wiman–Edge pencil. As an application, ...
Added: May 10, 2020
Loginov K., Moscow Mathematical Journal 2018 Vol. 18 No. 4 P. 721–737
We construct a standard birational model (a model that has Gorenstein canonical singularities) for the three-dimensional del Pezzo fibrations π: X→C of degree 1 and relative Picard number 1. We also embed the standard model into the relative weighted projective space ℙ_C(1,1,2,3). Our construction works in the G-equivariant category where G is a finite group. ...
Added: October 11, 2019
Perepechko A., Michałek M., Süß H., Mathematische Zeitschrift 2018 Vol. 290 No. 3-4 P. 1457–1478
We provide a new criterion for flexibility of affine cones over varieties covered by flexible affine varieties. We apply this criterion to prove flexibility of affine cones over secant varieties of Segre–Veronese embeddings and over certain Fano threefolds. We further prove flexibility of total coordinate spaces of Cox rings of del Pezzo surfaces. ...
Added: September 26, 2019
Perepechko A., Функциональный анализ и его приложения 2013 Т. 47 № 4 С. 45–52
We prove that the action of the special automorphism group on affine cones over del Pezzo surfaces of degree 4 and 5 is infinitely transitive. ...
Added: September 26, 2019
Serge Lvovski, Moscow Mathematical Journal 2019 Vol. 19 No. 3 P. 597–613
We show that if we are given a smooth non-isotrivial family of curves of genus 1 over C with a smooth base B for which the general fiber of the mapping J : B → A 1 (assigning j-invariant of the fiber to a point) is connected, then the monodromy group of the family (acting ...
Added: August 30, 2019
Trepalin A., / Series arXiv "math". 2017.
Let X be a minimal del Pezzo surface of degree 2 over a finite field 𝔽_q. The image Γ of the Galois group Gal(\bar{𝔽}_q/𝔽_q) in the group Aut(Pic(\bar{X})) is a cyclic subgroup of the Weyl group W(E_7). There are 60 conjugacy classes of cyclic subgroups in W(E_7) and 18 of them correspond to minimal del Pezzo surfaces. In this paper we study which possibilities of these subgroups for minimal del Pezzo ...
Added: December 2, 2018
Trepalin A., / Series arXiv "math". 2018.
Let X be a del Pezzo surface of degree 2 or greater over a finite field 𝔽_q. The image Γ of the Galois group Gal(\bar{𝔽}_q / 𝔽_q) in the group Aut(Pic(\bar{X})) is a cyclic subgroup preserving the anticanonical class and the intersection form. The conjugacy class of Γ in the subgroup of Aut(Pic(\bar{X})) preserving the anticanonical class and the intersection form is a natural invariant of X. We say that the ...
Added: December 2, 2018
Trepalin A., Loughran D., / Series arXiv "math". 2019.
We completely solve the inverse Galois problem for del Pezzo surfaces of degree 2 and 3 over all finite fields. ...
Added: December 2, 2018
Loginov K., / Series arXiv "math". 2018.
We consider threefold del Pezzo fibrations over a curve germ whose central fiber is non-rational. Under the additional assumption that the singularities of the total space are at worst ordinary double points, we apply a suitable base change and show that there is a 1-to-1 correpspondence between such fibrations and certain non-singular del Pezzo fibrations ...
Added: December 1, 2018
Cheltsov I., Shramov K., Park J., / Series math "arxiv.org". 2018.
We estimate δ-invariants of some singular del Pezzo surfaces with quotient singularities, which we studied ten years ago. As a result, we show that each of these surfaces admits an orbifold K\"ahler--Einstein metric. ...
Added: October 21, 2018
Trepalin A., Moscow Mathematical Journal 2018 Vol. 18 No. 3 P. 557–597
Let 𝕜 be any field of characteristic zero, X be a del Pezzo surface of degree 2 and G be a group acting on X. In this paper we study 𝕜-rationality questions for the quotient surface X/G. If there are no smooth 𝕜-points on X/G then X/G is obviously non-𝕜-rational. Assume that the set of smooth 𝕜-points on the quotient is not empty. We find ...
Added: October 4, 2018