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## DG quivers of smooth rational surfaces

Cornell University
,
2013.

Bodzenta-Skibinska A.

Let X be a smooth rational surface. We calculate a DG quiver of a full exceptional collection of line bundles on X obtained by an augmentation from a strong exceptional collection on the minimal model of X. In particular, we calculate canonical DG algebras of smooth toric surfaces.

Alexander I. Efimov, Journal of London Mathematical Society 2014 Vol. 90 No. 2 P. 350-372

In this paper, we construct infinitely many examples of toric Fano varieties with Picard number three, which do not admit full exceptional collections of line bundles. In particular, this disproves King's conjecture for toric Fano varieties. More generally, we prove that for any constant $c>\frac 34$ there exist infinitely many toric Fano varieties $Y$ with ...

Added: January 28, 2015

Shafarevich A., Results in Mathematics 2021 Vol. 76 No. 3 Article 145

Let KK be an algebraically closed field of characteristic zero and GaGa be the additive group of KK. We say that an irreducible algebraic variety X of dimension n over the field KK admits an additive action if there is a regular action of the group Gna=Ga×⋯×GaGan=Ga×⋯×Ga (n times) on X with an open orbit. In this paper we find all projective toric hypersurfaces admitting additive action. ...

Added: September 10, 2021

Efimov A., Journal of the European Mathematical Society 2020 Vol. 22 No. 9 P. 2879-2942

In this paper, we prove that the bounded derived category D-coh(b) (Y) of coherent sheaves on a separated scheme Y of finite type over a field k of characteristic zero is homotopically finitely presented. This confirms a conjecture of Kontsevich. We actually prove a stronger statement: D-coh(b) (Y) is equivalent to a DG quotient D-coh(b) ((Y) over tilde)/T, where (Y) over tilde is some smooth and proper ...

Added: September 24, 2020

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

Katzarkov L. V., Abouzaid M., Auroux D., Publications Mathématiques de l'IHÉS 2016 Vol. 123 No. 1 P. 199-282

https://link.springer.com/article/10.1007/s10240-016-0081-9 ...

Added: October 23, 2017

Zürich : European Mathematical Society Publishing house, 2012

The study of derived categories is a subject that attracts increasingly many young mathematicians from various fields of mathematics, including abstract algebra, algebraic geometry, representation theory and mathematical physics. The concept of the derived category of sheaves was invented by Grothendieck and Verdier in the 1960s as a tool to express important results in algebraic ...

Added: October 14, 2013

Ye L., / Cornell University. Series math "arxiv.org". 2020.

Thesis of the author. ...

Added: December 16, 2019

Kuznetsov A., Smirnov M., Belmans P., / Cornell University. Series arXiv "math". 2020.

For the derived category of the Cayley plane, which is the cominuscule Grassmannian of Dynkin type E_6, a full Lefschetz exceptional collection was constructed by Faenzi and Manivel. A general hyperplane section of the Cayley plane is the coadjoint Grassmannian of Dynkin type F_4. We show that the restriction of the Faenzi-Manivel collection to the hyperplane section ...

Added: August 19, 2020

Kuznetsov A., Shinder E., Karmazyn J., / Cornell University. Series arXiv "math". 2018.

We develop an approach that allows to construct semiorthogonal decompositions of derived categories of surfaces with rational singularities with components equivalent to derived categories of local finite dimensional algebras. First, we discuss how a semiorthogonal decomposition of a resolution of singularities of a surface X may induce a semiorthogonal decomposition of X. In the case when Xhas cyclic quotient ...

Added: December 3, 2018

Belmans P., Kuznetsov A., Smirnov M., Transformation Groups 2021

For the derived category of the Cayley plane, which is the cominuscule Grassmannian of Dynkin type E6, a full Lefschetz exceptional collection was constructed by Faenzi and Manivel. A general hyperplane section of the Cayley plane is the coadjoint Grassmannian of Dynkin type F4. We show that the restriction of the Faenzi–Manivel collection to such ...

Added: September 3, 2021

Shafarevich A., Moscow University Mathematics Bulletin 2019 Vol. 74 No. 5 P. 209-211

Let X be an affine toric variety over an algebraically closed field of characteristic zero. Orbits of connected component of identity of automorphism group in terms of dimensions of tangent spaces of the variety X are described. A formula to calculate these dimensions is presented. ...

Added: September 10, 2021

Katzarkov L. V., Deliu D., Favero D. et al., 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

Kuznetsov A., / Cornell University. Series arXiv "math". 2018.

We show that the derived category of a general Enriques surface can be realized as a semiorthogonal component in the derived category of a smooth Fano variety with a diagonal Hodge diamond. ...

Added: December 3, 2018

Galkin S., Shinder E., / Cornell University. Series math "arxiv.org". 2015. No. 1506.05831.

We define a zeta-function of a pre-triangulated dg-category and investigate its relationship with the motivic zeta-function in the geometric case. ...

Added: June 23, 2015

Galkin S., Katzarkov L. V., Mellit A. et al., Advances in Mathematics 2015 Vol. 278 P. 238-253

We conjecture that derived categories of coherent sheaves on fake projective n-spaces have a semi-orthogonal decomposition into a collection of n+1 exceptional objects and a category with vanishing Hochschild homology. We prove this for fake projective planes with non-abelian automorphism group (such as Keum’s surface). Then by passing to equivariant categories we construct new examples ...

Added: October 20, 2014

Feigin E., Makhlin I., / Cornell University. Series math "arxiv.org". 2020. No. 2008.13243.

We construct a family of flat semitoric degenerations for the Hibi variety of every finite distributive lattice. The irreducible components of each degeneration are the toric varieties associated with polytopes forming a regular subdivision of the order polytope of the underlying poset. These components are themselves Hibi varieties. For each degeneration in our family we ...

Added: September 1, 2020

Bodzenta-Skibinska A., Proceedings of the American Mathematical Society 2014

Bondal and Kapranov describe how to assign to a full exceptional collection on a variety X a DG category C such that the bounded derived category of coherent sheaves on X is equivalent to the bounded derived category of C. In this paper we show that the category C has finite dimensional spaces of morphisms. ...

Added: November 5, 2014

Galkin S., Belmans P., Mukhopadhyay S., Experimental Mathematics 2019

We give the first examples of smooth Fano and Calabi–Yau varieties violating the (narrow) canonical strip hypothesis, which concerns the location of the roots of Hilbert polynomials of polarized varieties. They are given by moduli spaces of rank 2 bundles with fixed odd-degree determinant on curves of sufficiently high genus, hence our Fano examples have ...

Added: October 4, 2019

Katzarkov L. V., Dimitrov G., Advances in Mathematics 2016 Vol. 288 P. 825-886

http://www.sciencedirect.com/science/article/pii/S0001870815004363 ...

Added: October 23, 2017

Fonarev A., Успехи математических наук 2014 Т. 69 № 4(418) С. 189-190

В работе приведена конструкция некоторых исключительных векторных расслоений на грассманианах, а также построено семейство полуортогональных разложений их ограниченных производных категорий. ...

Added: September 19, 2014

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