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## Fano-Mathieu correspondence

Cornell University
,
2018.
No. 1809.02738.

Galkin S.

We show that G-Fano threefolds are mirror-modular.
1. Mirror maps are inversed reversed Hauptmoduln for moonshine subgroups of SL2(ℝ).
2. Quantum periods, shifted by an integer constant (eigenvalue of quantum operator on primitive cohomology) are expansions of weight 2 modular forms (theta-functions) in terms of inversed Hauptmoduln.
3. Products of inversed Hauptmoduln with some fractional powers of shifted quantum periods are very nice cuspforms (eta-quotients).
The latter cuspforms also appear in work of Mason and others: they are eta-products, related to conjugacy classes of sporadic simple groups, such as Mathieu group M24 and Conway's group of isometries of Leech lattice.
This gives a strange correspondence between deformation classes of G-Fano threefolds and conjugacy classes of Mathieu group M24.

Galkin S., Rybakov S., / Cornell University. Series math "arxiv.org". 2019. No. 1910.14379.

For a family of K3 surfaces we implement a variation of a general construction of towers of algebraic curves over finite fields given in a previous paper. As a result we get a good tower over k=𝔽_{p^2}, that is optimal if p=3. ...

Added: November 6, 2019

Galkin S., / Cornell University. Series math "arxiv.org". 2018. No. 1809.02705.

We classify smooth Fano threefolds that admit degenerations to toric Fano threefolds with ordinary double points. ...

Added: September 25, 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

Gritsenko V., Никулин В. В., TRANSACTIONS OF THE MOSCOW MATHEMATICAL SOCIETY 2017 Т. 78 № 1 С. 89-100

Using our results about Lorentzian Kac--Moody algebras and arithmetic mirror symmetry, we give six series of examples of lattice-polarized K3 surfaces with automorphic discriminant. ...

Added: October 11, 2017

Kuznetsov A., Central European Journal of Mathematics 2012 Vol. 10 No. 4 P. 1198-1231

We introduce the notion of an instanton bundle on a Fano threefold of index 2. For such bundles we give an analogue of a monadic description and discuss the curve of jumping lines. The cases of threefolds of degree 5 and 4 are considered in a greater detail. ...

Added: September 10, 2012

Ivan Cheltsov, Karzhemanov I., Advances in Mathematics 2010 Vol. 223 P. 594-618

For any smooth quartic threefold in P4 we classify pencils on it whose general element is an irreducible surface birational to a surface of Kodaira dimension zero. ...

Added: December 6, 2013

Prokhorov Y., Sbornik Mathematics 2016 Vol. 207 No. 7 P. 983-1009

We study singular Fano threefolds of type $ V_{22}$. ...

Added: September 3, 2016

Guletski ̆i V., Tikhomirov A. S., Proceedings of the Edinburgh Mathematical Society 2016 Vol. 59 No. 02 P. 377-392

Let τ be the involution changing the sign of two coordinates in ℙ4. We prove that τ induces the identity action on the second Chow group of the intersection of a τ-invariant cubic with a τ-invariant quadric hypersurface in ℙ4. Let lτ andΠτ be the one- and two-dimensional components of the fixed locus of the involution τ. We describe the generalized Prymian associated with the ...

Added: December 28, 2015

Kiritchenko V., Tsfasman M., Vladuts S. et al., / Cornell University. Series arXiv "math". 2023.

Quadratic residue patterns modulo a prime are studied since 19th century. We state the last unpublished result of Lydia Goncharova, reformulate it and prior results in terms of algebraic geometry, and prove it. The core of this theorem is an unexpected relation between the number of points on a K3 surface and that on a ...

Added: December 3, 2023

Verbitsky M., Geometry and Topology 2014 Vol. 18 No. 2 P. 897-909

A Hermitian metric ω on a complex manifold is called SKT or pluriclosed if ddcω=0. Let M be a twistor space of a compact, anti-selfdual Riemannian manifold, admitting a pluriclosed Hermitian metric. We prove that in this case M is Kähler, hence isomorphic to CP3 or a flag space. This result is obtained from rational ...

Added: April 29, 2014

Cheltsov I., Shramov K., / Cornell University. Series math "arxiv.org". 2018.

We study the problem of existence of K\"ahler--Einstein metrics on smooth Fano threefolds of Picard rank one and anticanonical degree 22 that admit a faithful action of the multiplicative group ℂ∗. We prove that, except possibly two explicitly described cases, all such smooth Fano threefolds are K\"ahler--Einstein. ...

Added: October 21, 2018

Cheltsov I., Park J., Ahmadinezhad H., / Cornell University. Series arXiv "math". 2020.

We prove that a quasi-smooth Fano threefold hypersurface is birationally rigid if and only if it has Fano index one. ...

Added: August 19, 2020

Prokhorov Y., Kuznetsov A., / Cornell University. Series arXiv "math". 2017.

We give an explicit construction of prime Fano threefolds of genus 12 with a G_m-action, describe their isomorphism classes and automorphism groups. ...

Added: November 29, 2017

Zykin A. I., Contemporary Mathematics Series 2015 Vol. 637 P. 295-299

We prove under GRH that zeros of $L$-functions of modular forms of level $N$ and weight $k$ become uniformly distributed on the critical line when $N+k\to\infty.$ ...

Added: December 16, 2014

Cheltsov I., Przyjalkowski V., Shramov K., European Journal of Mathematics 2016 Vol. 2 No. 1 P. 96-119

We study quartic double solids admitting icosahedral symmetry. ...

Added: December 3, 2015

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

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

Kuznetsov A., Prokhorov Y., Shramov K., / Cornell University. Series arXiv "math". 2016.

We discuss various results on Hilbert schemes of lines and conics and automorphism groups of smooth Fano threefolds with 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: May 16, 2016

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

В. Л. Попов, Математические заметки 2017 Т. 102 № 1 С. 72-80

Мы доказываем, что аффинно-треугольные подгруппы являются борелевскими подгруппами групп Кремоны. ...

Added: May 3, 2017

Красноярск : ИВМ СО РАН, 2013

Труды Пятой Международной конференции «Системный анализ и информационные технологии» САИТ-2013 (19–25 сентября 2013 г., г.Красноярск, Россия): ...

Added: November 18, 2013

Min Namkung, Younghun K., Scientific Reports 2018 Vol. 8 No. 1 P. 16915-1-16915-18

Sequential state discrimination is a strategy for quantum state discrimination of a sender’s quantum
states when N receivers are separately located. In this report, we propose optical designs that can
perform sequential state discrimination of two coherent states. For this purpose, we consider not
only binary phase-shifting-key (BPSK) signals but also general coherent states, with arbitrary prior
probabilities. Since ...

Added: November 16, 2020

Grines V., Gurevich E., Pochinka O., Russian Mathematical Surveys 2017 Vol. 71 No. 6 P. 1146-1148

In the paper a Palis problem on finding sufficient conditions on embedding of Morse-Smale diffeomorphisms in topological flow is discussed. ...

Added: May 17, 2017