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## Betti tables of MCM modules over the cone of a plane cubic

Mathematische Zeitschrift. 2021. No. 297. P. 223-254.

We show that for maximal Cohen–Macaulay modules over the homogeneous coordinate ring of a smooth Calabi–Yau varieties *X*, the computation of Betti numbers can be reduced to computations of dimensions of certain HomHom spaces in the bounded derived category Db(X). In the simplest case of a smooth elliptic curve *E* embedded in P2 as a smooth cubic, we get explicit values for Betti numbers. The description of the automorphism group of the derived category Db(E) in terms of the spherical twist functors of Seidel and Thomas plays a major role in our approach. We show that there are only four possible shapes of the Betti tables up to shifts in internal degree, and two possible shapes up to shifts in internal degree and taking syzygies.

Proceedings of the American Mathematical Society 2020 Vol. 148 No. 4 P. 1373-1381

Let X be a smooth projective Calabi-Yau variety and let L be a Koszul line bundle on X. We show that for Betti numbers of a maximal Cohen-Macaulay module over the homogeneous coordinate ring A of X there are formulas similar to the formulas for cohomology numbers. This similarity is realized via the box-product resolution of ...

Added: October 31, 2020

Applied Mathematics and Computation 2014 Vol. 234 P. 380-384

We describe a simple implementation of the Takagi factorization of symmetric matrices $A = U\Lambda U^T$ with unitary $U$ and diagonal $\Lambda$ in terms of the square root of an auxiliary unitary matrix and the singular value decomposition of $A$. The method is based on an algebraically exact expression. For parameterized family $A_\epsilon = A ...

Added: June 4, 2014

European Journal of Mathematics 2018 Vol. 4 No. 2 P. 555-560

We construct pairs of elliptic curves over number fields with large intersection of projective torsion points. ...

Added: September 13, 2018

International Journal of Modern Physics A 2004 Vol. 19, May suppl. P. 418-435

Trigonometric degeneration of the Baxter-Belavin elliptic r matrix is described by the degeneration of the twisted WZW model on elliptic curves. The spaces of conformal blocks and conformal coinvariants of the degenerate model are factorised into those of the orbifold WZW model. ...

Added: August 14, 2014

On monodromy in families of elliptic curves over C / Cornell University. Series arXiv "math". 2018.

We show that if we are given a smooth non-isotrivial family of elliptic curves over ℂ with a smooth base B for which the general fiber of the mapping J:B→𝔸^1 (assigning j-invariant of the fiber to a point) is connected, then the monodromy group of the family (acting on H1(⋅,ℤ) of the fibers) coincides with SL(2,ℤ); if the general fiber has m≥2 connected components, then the ...

Added: December 5, 2018

Sharygin triangles and elliptic curves / Cornell University. Series math "arxiv.org". 2016.

The paper is devoted to the description of family of scalene triangles for which the triangle formed by the intersection points of bisectors with opposite sides is isosceles.
We call them Sharygin triangles.
It turns out that they are parametrized by an open subset of an elliptic curve.
Also we prove that there are infinitely many non-similar integer ...

Added: October 19, 2016

arXiv:1110.6917 Multiple Elliptic Polylogarithms Francis C. S. Brown, Andrey Levin / Cornell University. Series arXiv "math". 2013. No. 1110.6917 [.

Abstract. We study the de Rham fundamental group of the configuration sp ace of several marked points on a complex elliptic curve, and define multiple elliptic polylogarithms. These are multivalued functions with unipotent monodromy, and are constructed by a general averaging proce dure. We show that all iterated integrals on this configuration space can be ...

Added: October 4, 2013

Journal of Algebra 2019 Vol. 526 P. 211-242

We apply Orlov's equivalence to derive formulas for the Betti numbers of maximal Cohen-Macaulay modules over the cone an elliptic curve $(E,x)$ embedded into $\mathbb{P}^{n-1}$, by the full linear system $|\mathcal{O}(nx)|$, for $n>3$. The answers are given in terms of recursive sequences. These results are applied to give a criterion of (Co-)Koszulity.
In the last two ...

Added: May 24, 2019

On the Schlesinger transformation of isomonodromic families over elliptic curve / ИТЭФ. Series "Препринты ИТЭФ". 2012. No. 57/12.

In this work we investigate the action of generalized Schlesinger transformation on the isomonodromic families of meromorphic connections on the linear bundles of rank two and degree zero over an elliptic curve. The main interest is the action of the gauge transformation on the moduli space of vector bundles. the central result is the explicit ...

Added: March 31, 2014

Математические вопросы криптографии 2014 Vol. 5 No. 2 P. 99-102

In this article we present an algorithm for constructing an elliptic curve endomorphism for given complex irrationality. This endomorphism can be used for speeding up a group operation on elliptic curve. ...

Added: February 2, 2015

Mathematical notes 2016 Vol. 99 No. 3 P. 397-405

Let S be a bielliptic surface over a finite field, and let an elliptic curve B be the Albanese variety of S; then the zeta function of the surface S is equal to the zeta function of the direct product P1 × B. Therefore, the classification problem for the zeta functions of bielliptic surfaces is ...

Added: July 8, 2016

Advances in Mathematics 2016 Vol. 295 P. 195-249

Building upon ideas of Eisenbud, Buchweitz, Positselski, and others, we introduce the notion of a factorization category. We then develop some essential tools for working with factorization categories, including constructions of resolutions of factorizations from resolutions of their components and derived functors. Using these resolutions, we lift fully-faithfulness and equivalence statements from derived categories of ...

Added: October 23, 2017

Proceedings of the American Mathematical Society 2014 Vol. 142 P. 15-19

We show that neither the Barvinok rank nor the Kapranov rank of a tropical matrix M can be defined in terms of the regular mixed subdivision produced by M. This answers a question asked by Develin, Santos and Sturmfels. ...

Added: October 5, 2013

Arnold Mathematical Journal 2019 Vol. 5 No. 1 P. 23-35

We present an improved construction of the fundamental matrix factorization in the FJRW-theory given in Polishchuk and Vaintrob (J Reine Angew Math 714:1—22, 2016). The revised construction is coordinate-free and works for a possibly nonabelian finite group of symmetries. One of the new ingrediants is the category of dg-matrix factorizations over a dg-scheme. ...

Added: September 4, 2019

arXiv:1110.6917v2 [math.NT] 20 Jun 2013 MULTIPLE ELLIPTIC POLYLOGARITHM / . 2013. No. 1110.6917.

Abstract. We study the de Rham fundamental group of the configuration space
E (n) of n + 1 marked points on a complex elliptic curve E, and define multiple
elliptic polylogarithms. These are multivalued functions on E (n) with unipotent
monodromy, and are constructed by a general averaging procedure. We show
that all iterated integrals on E (n) , ...

Added: May 14, 2014

Journal of Physics A: Mathematical and Theoretical 2001 Vol. 34 No. 11 P. 2403-2413

We construct a Gaudin type lattice model as the Wess-Zumino-Witten model on elliptic curves at the critical level. Bethe eigenvectors are obtained by the bosonisation technique. ...

Added: August 14, 2014

MULTIPLE ELLIPTIC POLYLOGARITHM / . 2013. No. arXiv:1110.6917v2 .

Abstract. We study the de Rham fundamental group of the configuration space
E (n) of n + 1 marked points on a complex elliptic curve E, and define multiple
elliptic polylogarithms. These are multivalued functions on E (n) with unipotent
monodromy, and are constructed by a general averaging procedure. We show
that all iterated integrals on E (n) , ...

Added: May 14, 2014

Selecta Mathematica, New Series 2018 Vol. 24 No. 1 P. 21-62

We study plane partitions satisfying condition a_{n+1,m+1}=0 (this condition is called “pit”) and asymptotic conditions along three coordinate axes. We find the formulas for generating function of such plane partitions. Such plane partitions label the basis vectors in certain representations of quantum toroidal gl1 algebra, therefore our formulas can be interpreted as the characters of ...

Added: October 24, 2018

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

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

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

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

Moscow Mathematical Journal 2017 Vol. 17 No. 4 P. 565-600

We associate an explicit equivalent descendent insertion to any relative insertion in quantum K-theory of Nakajima varieties.
This also serves as an explicit formula for off-shell Bethe eigenfunctions for general quantum loop algebras associated to quivers and gives the general integral solution to the corresponding quantum Knizhnik Zamolodchikov and dynamical q-difference equations. ...

Added: October 25, 2018