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Group actions on affine cones
P. 123–163.
In this article, the authors study the action of the additive group C on affine cones over projective varieties. They show that such actions always exist for the cones over del Pezzo surfaces of degree d≥4 which are canonically embedded, and give relations between the actions and existence of polar cylinders. The case of del Pezzo surfaces of degree 3 is still open; for example, it is not known if the variety of the equation w3+x3+y3+z3=0 in C4 admits an action of the additive group C.
In book
Vol. 54: Affine Algebraic Geometry: The Russell Festschrift. , Providence: American Mathematical Society, 2011.
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
Loginov K., Przyjalkowski V., Trepalin A., Труды Математического института им. В.А. Стеклова РАН 2025 Т. 329 С. 132–164
We introduce and study the notion of G-coregularity of algebraic varieties endowed with an action of a finite group G. We compute the G-coregularity of smooth del Pezzo surfaces of degree at least 6, and give a characterization of groups that can act on conic bundles with G-coregularity 0. We describe the relations between the notions of G-coregularity, G-log ...
Added: September 4, 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
Golota A., Известия РАН. Серия математическая 2024 Т. 88 № 5 С. 47–66
Let X be a complex projective variety. Suppose that the group of birational automorphisms of X contains finite subgroups isomorphic to (Z/NZ)^r for r fixed and N arbitrarily large. We show that r does not exceed 2dim(X). Moreover, the equality holds if and only if X is birational to an abelian variety. We also show that an analogous ...
Added: November 6, 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
Prokhorov Y., Zaidenberg M., , in: The Art of Doing Algebraic Geometry.: Birkhäuser, 2023. P. 363–383.
We show that the affine cones over any Fano–Mukai fourfold of genus 10 are flexible in the sense of [1]. In particular, the automorphism group of such a cone acts highly transitively outside the vertex. Furthermore, any Fano–Mukai fourfold of genus 10, with one exception, admits a covering by open charts isomorphic to ...
Added: November 13, 2023
Kuznetsov A., Prokhorov Y., American Journal of Mathematics 2023 Vol. 145 No. 2 P. 335–411
We give necessary and sufficient conditions for unirationality and rationality of Fano threefolds of geometric Picard rank-1 over an arbitrary field of zero characteristic. ...
Added: September 1, 2023
Prokhorov Y., Rendiconti del Circolo Matematico di Palermo 2023 Vol. 2 No. 72 P. 1797–1821
We classify nonrational Fano threefolds X with terminal Gorenstein singularities such that $\mathrm{\rk}\, \mathrm{\Pic}(X)=1$, (−KX)3≥8, and $\mathrm{\rk}\, \mathrm{\Cl}(X)\le 2$. ...
Added: September 1, 2023
Cham: Springer, 2023.
This book collects the proceedings of a series of conferences dedicated to birational geometry of Fano varieties held in Moscow, Shanghai and Pohang
The conferences were focused on the following two related problems:
• existence of Kähler–Einstein metrics on Fano varieties
• degenerations of Fano varieties
on which two famous conjectures were recently proved. The first is the famous ...
Added: May 24, 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
Alexander Kuznetsov, Prokhorov Y., Journal of the Institute of Mathematics of Jussieu 2024 Vol. 23 No. 1 P. 207–247
We prove rationality criteria over nonclosed fields of characteristic 00 for five out of six types of geometrically rational Fano threefolds of Picard number 11 and geometric Picard number bigger than 11 . 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. ...
Added: November 30, 2022
Prokhorov Y., ELECTRONIC RESEARCH ARCHIVE 2022 Vol. 30 No. 5 P. 1881–1897
We obtain a sufficient condition for a Fano threefold with terminal singularities to have a conic bundle structure. ...
Added: November 28, 2022
Vikulova A., / Series arXiv "math". 2022.
In this paper we prove that for n-dimensional smooth l-Fano well formed weighted complete intersections, which is not isomorphic to a usual projective space, the upper bound for l is equal to ⌈log2(n+2)⌉−1. We also prove that the only l-Fano of dimension n among such manifolds with inequalities ⌈log3(n+2)⌉⩽l⩽⌈log2(n+2)⌉−1 is a complete intersection of quadrics in a usual projective space. ...
Added: November 27, 2022
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
Loginov K., European Journal of Mathematics 2021 Vol. 8 No. 3 P. 991–1005
Consider a family of Fano varieties π:X→B∋o over a curve germ with a smooth total space X. Assume that the generic fiber is smooth and the special fiber F=π−1(o) has simple normal crossings. Then F is called a semistable degeneration of Fano varieties. We show that the dual complex of F is a simplex of dimension not greater than dim F. Simplices ...
Added: September 3, 2021
Perepechko A., Forum Mathematicum 2021 Vol. 33 No. 2 P. 339–348
Let Y be a smooth del Pezzo surface of degree 3 polarized by a very ample divisor that is not proportional to the anticanonical one. Then the affine cone over Y is flexible in codimension one. Equivalently, such a cone has an open subset with an infinitely transitive action of the special automorphism group on ...
Added: January 15, 2021