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## 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