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Of all publications in the section: 482
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Working paper
Alexeev V., Orlov D. O. arxiv.org. math. Cornell University, 2012
Working paper
Alexander Kuznetsov, Perry A. arxiv.org. math. Cornell University, 2014
Given a variety Y with a rectangular Lefschetz decomposition of its derived category, we consider a degree n cyclic cover X→Y ramified over a divisor Z⊂Y. We construct semiorthogonal decompositions of D^b(X) and D^b(Z) with distinguished components A_X and A_Z, and prove the equivariant category of A_X (with respect to an action of the n-th roots of unity) admits a semiorthogonal decomposition into n−1 copies of A_Z.
Working paper
Alexander I. Efimov. arxiv.org. math. Cornell University, 2014
In this paper we study the derived categories of coherent sheaves on Grassmannians Gr(k,n), defined over the ring of integers. We prove that the category D^b(Gr(k,n)) has a semi-orthogonal decomposition, with components being full subcategories of the derived category of representations of GL_k. This in particular implies existence of a full exceptional collection, which is a refinement of Kapranov's collection \cite{Kap}, which was constructed over a field of characteristic zero. We also describe the right dual semi-orthogonal decomposition which has a similar form, and its components are full subcategories of the derived category of representations of GL_{n−k}. The resulting equivalences between the components of the two decompositions are given by a version of Koszul duality for strict polynomial functors. We also construct a tilting vector bundle on Gr(k,n). We show that its endomorphism algebra has two natural structures of a split quasi-hereditary algebra over Z, and we identify the objects of D^b(Gr(k,n)), which correspond to the standard and costandard modules in both structures. All the results automatically extend to the case of arbitrary commutative base ring and the category of perfect complexes on the Grassmannian, by extension of scalars (base change). Similar results over fields of arbitrary characteristic were obtained independently in \cite{BLVdB}, by different methods.
Working paper
Kuznetsov A. arxiv.org. math. Cornell University, 2015
We discuss a relation between the structure of derived categories of smooth projective varieties and their birational properties. We suggest a possible definition of a birational invariant, the derived category analogue of the intermediate Jacobian, and discuss its possible applications to the geometry of prime Fano threefolds and cubic fourfolds.
Working paper
Balzin Edouard. arxiv.org. math. Cornell University, 2014
In the world of triangulated categories, categorical resolutions (as defined by Kuznetsov and Luntz) have been useful. One would like to have a similar notion of categorical resolution in homotopical algebra, where one works with categories which are not additive, such as the categories of E_n-algebras. Describing algebraic structures using the approach inspired by Segal, we transfer the question to the setting of homotopical Grothendieck (op)fibrations. We then introduce the notion of derived section of a homotopical Grothendieck (op)fibration, and show that a selected class of functors gives rise to (partial) categorical resolutions for homotopical categories of derived sections.
Working paper
Elagin A. D. arxiv.org. math. Cornell University, 2012. No. 1206.2881.

In this paper a method of constructing a semiorthogonal decomposition of the derived category of G-equivariant sheaves on a variety X is described, provided that the derived category of sheaves on X admits a semiorthogonal decomposition, whose components are preserved by the action of the group G on X. Using this method, semiorthogonal decompositions of equivariant derived categories were obtained for projective bundles and for blow-ups with a smooth center, and also for varieties with a full exceptional collection, preserved by the action of the group. As a main technical instrument, descent theory for derived categories is used.

Working paper
Feigin E., Cerulli Irelli G., Reineke M. arxiv.org. math. Cornell University, 2012. No. 1209.3960.

A desingularization of arbitrary quiver Grassmannians for representations of Dynkin quivers is constructed in terms of quiver Grassmannians for an algebra derived equivalent to the Auslander algebra of the quiver.

Working paper
Rybakov S. arxiv.org. math. Cornell University, 2013
For a morphism of smooth schemes over a regular affine base we define functors of derived direct image and extraordinary inverse image on coderived categories of DG-modules over de Rham DG-algebras. Positselski proved that for a smooth algebraic variety $X$ over a field $k$ of characteristic zero the coderived category of DG-modules over $\Omega^\bullet_{X/k}$ is equivalent to the unbounded derived category of quasi-coherent right $\DD_X$-modules. We prove that our functors correspond to the functors of the same name for $\DD_X$-modules under Positselski equivalence.
Working paper
Bodzenta-Skibinska A. arxiv.org. math. Cornell University, 2013
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.
Working paper
Khoroshkin A., Willwacher T., Živković M. arxiv.org. math. Cornell University, 2014. No. 1411.2369.
We study the cohomology of complexes of ordinary (non-decorated) graphs, introduced by M. Kontsevich. We construct spectral sequences converging to zero whose first page contains the graph cohomology. In particular, these series may be used to show the existence of an infinite series of previously unknown and provably non-trivial cohomology classes, and put constraints on the structure of the graph cohomology as a whole.
Working paper
Buchstaber V., Limonchenko I. arxiv.org. math. Cornell University, 2018. No. 1811.02221.
We develop a theory of direct families of polytopes with nontrivial Massey products
Working paper
Takebe T. arxiv.org. math. Cornell University, 2013. No. 1308.4584.
We show that N-variable reduction of the dispersionless BKP hierarchy is described by a Loewner type equation for the quadrant.
Working paper
Timorin V. arxiv.org. math. Cornell University, 2013. No. 1309.4879.
Section 235 of Gauss' fundamental treatise "Disquisitiones Arithmeticae" establishes basic properties that compositions of binary quadratic forms must satisfy. Although this section is very technical, it contains truly important results. We review section 235 using a more invariant language and simplifying the arguments. We also make the statements slightly stronger by removing unnecessary assumptions.
Working paper
Valentina Kiritchenko. arxiv.org. math. Cornell University, 2013. No. 1307.7234.
We define convex-geometric counterparts of divided difference (or Demazure) operators from the Schubert calculus and representation theory. These operators are used to construct inductively polytopes that capture Demazure characters of representations of reductive groups. In particular, Gelfand-Zetlin polytopes and twisted cubes of Grossberg-Karshon are obtained in a uniform way. This preprint contains the proofs of results announced in Oberwolfach Reports in the talk with the same title.
Working paper
Gorinov A. arxiv.org. math. Cornell University, 2005. No. 0511593.
The main purpose of this paper is to show that the mixed Hodge polynomial of the space of equations'' for smooth complete intersections of given multidegree in $\mathbb{C} P^n$ the quotient being the mixed Hodge polynomial of the corresponding quotient space. As a by-product of the method used in the proof, we obtain expressions divisible by the order the automorphism group of any smooth projective hypersurface of given dimension and degree.
Working paper
Gorinov A. arxiv.org. math. Cornell University, 2013. No.  arXiv:1303.5150 [math.AG].
In this note we extend some of the results of a previous paper \url{arXiv:math/0511593} to algebraically closed fields of finite characteristic. In particular, we show that there is an explicit expression in $n$ and $d$ which is divisible by the prime to $p$ part of the order of the the automorphism group of a smooth degree $d$ hypersurface $\subset \mathbb{P}^n_k$ for $k$ an algebraically closed field of characteristic $p$.
Working paper
Victor Kulikov, Shustin E. arxiv.org. math. Cornell University, 2014
We study the geometry of equiclassical strata of the discriminant in the space of plane curves of a given degree, which are families of curves of given degree, genus and class (degree of the dual curve). Our main observation is that the use of duality transformation leads to a series of new sufficient conditions for a regular behavior of the equiclassical stratification. We also discuss duality of curves in higher-dimensional projective spaces and in Grassmannians with focus on similar questions of the regularity of equiclassical families of spacial curves.
Working paper
Kochetkov Y. arxiv.org. math. Cornell University, 2019. No. 1911.09321.
We consider quadrangles of perimeter 2 in the plane with marked directed edge. To such quadrangle $Q$ a two-dimensional plane $\Pi\in\mathbb{R}^4$ with orthonormal base is corresponded. Orthogonal plane $\Pi^bot$ defines a plane quadrangle $Q^\circ$ of perimeter 2 and with marked directed edge. This quadrangle is defined uniquely (up to rotation and symmetry). Quadrangles $Q$ and $Q^\circ$ will be called dual to each other. The following properties of duality are proved: a) duality preserves convexity, non convexity and self-intersection; b) duality preserves the length of diagonals; c) the sum of lengths of corresponding edges in $Q$ and $Q^\circ$ is 1.