### ?

## Delta invariants of smooth cubic surfaces

European Journal of Mathematics. 2019. Vol. 5. P. 729-762.

Cheltsov I., Zhang K.

We prove that 𝛿δ-invariants of smooth cubic surfaces are at least 6/5.

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

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

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

Prokhorov Y., / Cornell University. Series arXiv "math". 2015. No. 1508.04371.

We study singular Fano threefolds of type V22. ...

Added: October 9, 2015

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

Cheltsov I., Park J., Shramov K., Journal of Geometric Analysis 2020

We use the methods introduced by Cheltsov–Rubinstein–Zhang (Sel Math (N.S.) 25(2):25–34, 2019) to estimate δ-invariants of the seven singular del Pezzo surfaces with quotient singularities studied by Cheltsov–Park–Shramov (J Geom Anal 20(4):787–816, 2010) that have α-invariants less than 2/3. As a result, we verify that each of these surfaces admits an orbifold Kähler–Einstein metric. ...

Added: May 10, 2020

Ю. Г. Прохоров, Известия РАН. Серия математическая 2013 Т. 77 № 3 С. 199-222

We study elements $\tau$ of order two in the birational automorphism groups of rationally connected three-dimensional algebraic varieties such that there exists a non-uniruled divisorial component of the $\tau$-fixed point locus. Using the equivariant minimal model program, we give a rough classification of such elements. ...

Added: July 1, 2013

Aleksei Golota, / Cornell University. 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

Loginov K., / Cornell University. 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

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

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

Prokhorov Y., Advances in Geometry 2013 Vol. 13 No. 3 P. 419-434

We classify Fano threefolds with only Gorenstein terminal singularities and Picard
number greater than 1, satisfying the additional assumption that the G-invariant part of the Weil
divisor class group is of rank 1 with respect to an action of some group G. ...

Added: October 7, 2013

Shramov K., / Cornell University. Series arXiv "math". 2020.

We classify finite groups acting by birational transformations of a non-trivial Severi--Brauer surface over a field of characteristc zero that are not conjugate to subgroups of the automorphism group. Also, we show that the automorphism group of a smooth cubic surface over a field K of characteristic zero that has no K-points is abelian, and find a sharp ...

Added: August 19, 2020

Prokhorov Y., Sbornik Mathematics 2013 Vol. 204 No. 3 P. 347-382

We classify $\mathbb Q$-Fano threefolds of Fano index > 2 and sufficiently big degree. ...

Added: October 7, 2013

Kishimoto T., Yuri Prokhorov, Zaidenberg M., Osaka Journal of Mathematics 2014 Vol. 51 No. 4 P. 1093-1113

We address the following question: When an affine cone over a smooth Fano threefold admits an effective action of the additive group? In this paper we deal with Fano threefolds of index 1 and Picard number 1. Our approach is based on a geometric criterion from our previous paper, which relates the existence of an ...

Added: October 10, 2013

Vikulova A., / Cornell University. 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

Cheltsov Ivan, Wilson A., Journal of Geometric Analysis 2013 Vol. 23 No. 3 P. 1257-1289

We classify smooth del Pezzo surfaces whose α-invariant of Tian is bigger than 1. ...

Added: November 14, 2013

Cheltsov I., Zhang K., / Cornell University. Series arXiv "math". 2018.

We prove that δ-invariants of smooth cubic surfaces are at least 6/5. ...

Added: December 3, 2018

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., Advances in Geometry 2013 Vol. 13 No. 3 P. 389-418

We classify Fano threefolds with only terminal singularities whose canonical class is
Cartier and divisible by 2 with the additional assumption that the G-invariant part of the Weil divisor
class group is of rank 1 with respect to an action of some group G. In particular, we find a lot of
examples of Fano 3-folds with “many” symmetries. ...

Added: October 7, 2013

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., Cheltsov I., Zaidenberg M. et al., / Cornell University. Series arXiv "math". 2020.

This paper is a survey about cylinders in Fano varieties and related problems. ...

Added: August 19, 2020

Galkin S., Shinder E., / Cornell University. 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

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