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Affine Cones over Fano–Mukai Fourfolds of Genus 10 are Flexible
P. 363–383.
Prokhorov Y., Zaidenberg M.
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
Keywords: Fano variety
Publication based on the results of:
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
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
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
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
Przyjalkowski V., Shramov K., Communications in Number Theory and Physics 2020 Vol. 14 No. 3 P. 511–553
We prove that if a smooth variety with non-positive canonical class can be embedded into a weighted projective space of dimension n as a well formed complete intersection and it is not an intersection with a linear cone therein, then the weights of the weighted projective space do not exceed n+1. Based on this bound ...
Added: October 13, 2020
Prokhorov Y., Cheltsov I., Zaidenberg M. et al., / Series arXiv "math". 2020.
This paper is a survey about cylinders in Fano varieties and related problems. ...
Added: August 19, 2020
Cheltsov I., Zhang K., European Journal of Mathematics 2019 Vol. 5 P. 729–762
We prove that 𝛿δ-invariants of smooth cubic surfaces are at least 6/5. ...
Added: May 10, 2020
Loginov K., / Series arXiv "math". 2019.
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 ≤dim F. Simplices of any admissible dimension can be realized ...
Added: October 11, 2019
Aleksei Golota, / Series arXiv "math". 2019.
For a polarized variety (X,L) and a closed connected subgroup G⊂Aut(X,L) we define a G-invariant version of the δ-threshold. We prove that for a Fano variety (X,−KX) and a connected subgroup G⊂Aut(X) this invariant characterizes G-equivariant uniform K-stability. We also use this invariant to investigate G-equivariant K-stability of some Fano varieties with large groups of ...
Added: October 7, 2019
Cheltsov I., Dubouloz A., Park J., Compositio Mathematica 2018 Vol. 154 No. 11 P. 2462–2484
We study a wide class of affine varieties, which we call affine Fano varieties. By analogy with birationally super-rigid Fano varieties, we define super-rigidity for affine Fano varieties, and provide many examples and non-examples of super-rigid affine Fano varieties. ...
Added: October 17, 2018
Prokhorov Y., Zaidenberg M., European Journal of Mathematics 2018 Vol. 4 No. 3 P. 1197–1263
It is known that the moduli space of smooth Fano–Mukai fourfolds V18 of genus 10 has dimension one. We show that any such fourfold is a completion of ℂ4 in two different ways. Up to isomorphism, there is a unique fourfold Vs18 acted upon by SL2(ℂ). The group Open image in new window is a ...
Added: September 6, 2018
Prokhorov Y., Zaidenberg M., European Journal of Mathematics 2016 Vol. 2 No. 1 P. 262–282
We construct four different families of smooth Fano fourfolds with Picard rank 1, which contain cylinders, i.e., Zariski open subsets of the form Z ×A1, where Z is a quasiprojective variety. The affine cones over such a fourfold admit effective Ga-actions. Similar constructions of cylindrical Fano threefolds were done previously in the papers by Kishimoto ...
Added: November 27, 2015
Bondal A. I., Zhdanovskiy I., , in: Primitive Forms and Related Subjects — Kavli IPMU 2014.: Tokyo: Mathematical Society of Japan, 2019. P. 1–18.
The goal of these notes is to show that the classification problem of algebraically unbiased system of projectors has an interpretation in symplectic geometry. This leads us to a description of the moduli space of algebraically unbiased bases as critical points of a potential functions, which is a Laurent polynomial in suitable coordinates. The Newton ...
Added: October 22, 2015
Prokhorov Y., / Series arXiv "math". 2015. No. 1508.04371.
We study singular Fano threefolds of type V22. ...
Added: October 9, 2015
Galkin S., Shinder E., / Series math "arxiv.org". 2014. No. 1405.5154.
We find a relation between a cubic hypersurface Y and its Fano variety of lines F(Y) in the Grothendieck ring of varieties. We prove that if the class of an affine line is not a zero-divisor in the Grothendieck ring of varieties, then Fano variety of lines on a smooth rational cubic fourfold is birational ...
Added: May 21, 2014
Cheltsov Ivan, Shramov Constantin, Experimental Mathematics 2013 Vol. 22 No. 3 P. 313–326
We study del Pezzo surfaces that are quasismooth and well-formed weighted hypersurfaces. In particular, we find all such surfaces whose α-invariant of Tian is greater than 2/3. ...
Added: January 27, 2014
Prokhorov Y., Springer Proceedings in Mathematics & Statistics 2014 Vol. 79 P. 215–229
We give a sharp bound for orders of elementary abelian 2-groups of birational automorphisms of rationally connected threefolds. ...
Added: January 24, 2014
Yuri Prokhorov, Documenta Mathematica 2010 Vol. 15 P. 843–872
We study Q-Fano threefolds of large Fano index. In
particular, we prove that the maximum possible Fano index is attained
only by the weighted projective space P(3,4,5,7). ...
Added: December 6, 2013
Cheltsov Ivan, Park J., Won J., Mathematische Zeitschrift 2014 No. 276 P. 51–79
We study log canonical thresholds on quartic threefolds, quintic fourfolds, and double spaces. As an important application, we show that they have Kähler–Einstein metrics if they are general. ...
Added: November 14, 2013