### ?

## G-Fano threefolds, I

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.

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

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

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

Added: October 9, 2015

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

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

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

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

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

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

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

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., 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

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

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

Coates T., Galkin S., Kasprzyk A. et al., / Cornell University. Series math "arxiv.org". 2014. No. 1406.4891.

We collect a list of known four-dimensional Fano manifolds and compute their quantum periods. This list includes all four-dimensional Fano manifolds of index greater than one, all four-dimensional toric Fano manifolds, all four-dimensional products of lower-dimensional Fano manifolds, and certain complete intersections in projective bundles. ...

Added: June 20, 2014

Galkin S., Golyshev V., Iritani H., / Cornell University. Series math "arxiv.org". 2014. No. 1404.6407.

We propose Gamma Conjectures for Fano manifolds which can be thought of as a square root of the index theorem. Studying the exponential asymptotics of solutions to the quantum differential equation, we associate a principal asymptotic class A_F to a Fano manifold F. We say that F satisfies Gamma Conjecture I if A_F equals the ...

Added: May 4, 2014

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

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

Akhtar M., Coates T., Galkin S. et al., Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 2012 Vol. 8 No. 094 P. 1–707

Given a Laurent polynomial f, one can form the period of f: this is a function of one complex variable that plays an important role in mirror symmetry for Fano manifolds. Mutations are a particular class of birational transformations acting on Laurent polynomials in two variables; they preserve the period and are closely connected with ...

Added: September 14, 2013

Galkin S., / Cornell University. Series math "arxiv.org". 2014. No. 1404.7388.

Consider a Laurent polynomial with real positive coefficients such that the origin is strictly inside its Newton polytope. Then it is strongly convex as a function of real positive argument. So it has a distinguished Morse critical point --- the unique critical point with real positive coordinates. As a consequence we obtain a positive answer ...

Added: May 4, 2014

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

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

Prokhorov Y., Kuznetsov A., Shramov K., Japanese Journal of Mathematics 2018 Vol. 13 No. 1 P. 109–185

We discuss various results on Hilbert schemes of lines and conics and automorphism groups of smooth Fano threefolds of Picard rank 1. Besides a general review of facts well known to experts, the paper contains some new results, for instance, we give a description of the Hilbert scheme of conics on any smooth Fano threefold ...

Added: November 22, 2017

Prokhorov Y., Mori S., Известия РАН. Серия математическая 2019 Т. 83 № 3 С. 158–212

Росток экстремальной окрестности – это аналитический росток трехмерного многообразия с терминальными особенностями вдоль приведенной полной кривой, допускающий стягивание, слои которого не более чем одномерны. Цель настоящей статьи – дать обзор результатов, касающихся стягиваний с неприводимым центральным слоем, содержащих только одну негоренштейнову точку. ...

Added: June 4, 2019