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New examples of cylindrical Fano fourfolds
Cornell University
,
2015.
No. 1507.01748.
Prokhorov Y., Zaidenberg M.
We construct new families of smooth Fano fourfolds with Picard rank 1, which contain cylinders, i.e., Zariski open subsets of form Z×, where Z is a quasiprojective variety. The affine cones over such a fourfold admit effective G_a-actions. Similar constructions of cylindrical Fano threefolds and fourfolds were done previously in [KPZ11, KPZ14, PZ15].
Publication based on the results of:
Coates T., Galkin S., Kasprzyk A. et al., Experimental Mathematics 2020 Vol. 29 No. 2 P. 183-221
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: September 1, 2018
Coates T., Corti A., Galkin S. et al., Geometry and Topology 2016 Vol. 20 No. 1 P. 103-256
The quantum period of a variety X is a generating function for certain Gromov-Witten invariants of X which plays an important role in mirror symmetry. In this paper we compute the quantum periods of all 3-dimensional Fano manifolds. In particular we show that 3-dimensional Fano manifolds with very ample anticanonical bundle have mirrors given by ...
Added: November 18, 2014
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
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
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
Galkin S., / Cornell University. Series math "arxiv.org". 2016. No. 1604.04652.
For Fano manifolds we define Ap\'ery constants and Ap\'ery class as particular limits of ratios of coefficients of solutions of the quantum differential equation. We do numerical computations in case of homogeneous varieties. These numbers are identified to be polynomials in the values of Riemann zeta-function with natural arguments. ...
Added: April 19, 2016
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
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., 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
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
Galkin S., Iritani H., / Cornell University. Series math "arxiv.org". 2015. No. 1508.00719.
The asymptotic behaviour of solutions to the quantum differential equation of a Fano manifold F defines a characteristic class A_F of F, called the principal asymptotic class. Gamma conjecture of Vasily Golyshev and the present authors claims that the principal asymptotic class A_F equals the Gamma class G_F associated to Euler's Γ-function. We illustrate in ...
Added: August 5, 2015
Coates T., Corti A., Galkin S. et al., / Cornell University. Series math "arxiv.org". 2012. No. 1212.1722.
We consider mirror symmetry for Fano manifolds, and describe how one can recover the classification of 3-dimensional Fano manifolds from the study of their mirrors. We sketch a program to classify 4-dimensional Fano manifolds using these ideas. ...
Added: September 14, 2013
Kuznetsov A., Известия РАН. Серия математическая 2019 Т. 83 № 3 С. 127-132
Мы показываем, что ограниченная производная категория когерентных пучков на общей поверхности Энриквеса может быть реализована как полуортогональная компонента в производной категории гладкого многообразия Фано с диагональным ромбом Ходжа. ...
Added: June 4, 2019
Galkin S., Golyshev V., Iritani H., Duke Mathematical Journal 2016 Vol. 165 No. 11 P. 2005-2077
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: November 18, 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., 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
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
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
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
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 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
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
Kotelnikova M. V., Aistov A., Вестник Нижегородского университета им. Н.И. Лобачевского. Серия: Социальные науки 2019 Т. 55 № 3 С. 183-189
The article describes a method that allows to improve the content of disciplines of the mathematical cycle by dividing them into invariant (general) and variable parts. The invariants were identified for such disciplines as «Linear algebra», «Mathematical analysis», «Probability theory and mathematical statistics» delivered to Bachelors program students of economics at several universities. Based on ...
Added: January 28, 2020
Borzykh D., ЛЕНАНД, 2021
Книга представляет собой экспресс-курс по теории вероятностей в контексте начального курса эконометрики. В курсе в максимально доступной форме изложен тот минимум, который необходим для осознанного изучения начального курса эконометрики. Данная книга может не только помочь ликвидировать пробелы в знаниях по теории вероятностей, но и позволить в первом приближении выучить предмет «с нуля». При этом, благодаря доступности изложения и небольшому объему книги, ...
Added: February 20, 2021