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Alexander Kuznetsov

Publications and source records attributed to Alexander Kuznetsov.

At least 55 records · Page 3Linked to original sources
arXiv2019

On residual categories for Grassmannians

We define and discuss some general properties of residual categories of Lefschetz decompositions in triangulated categories. In the case of the derived category of coherent sheaves on the Grassmannian $\text{G}(k,n)$ we conjecture that the residual category associated with Fonarev's Lefschetz exceptional collection is generated by a completely orthogonal exceptional collection. We prove this conjecture for $k = p$, a prime number, modulo completeness of Fonarev's collection (and for $p = 3$ we check this completeness).

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arXiv2019

Categorical cones and quadratic homological projective duality

We introduce the notion of a categorical cone, which provides a categorification of the classical cone over a projective variety, and use our work on categorical joins to describe its behavior under homological projective duality. In particular, our construction provides well-behaved categorical resolutions of singular quadrics, which we use to obtain an explicit quadratic version of the main theorem of homological projective duality. As applications, we prove the duality conjecture for Gushel-Mukai varieties, and produce interesting examples of conifold transitions between noncommutative and honest Calabi-Yau threefolds.

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arXiv2018

Gushel-Mukai varieties: moduli

We describe the moduli stack of Gushel-Mukai varieties as a global quotient stack and its coarse moduli space as the corresponding GIT quotient. The construction is based on a comprehensive study of the relation between this stack and the stack of Lagrangian data; roughly speaking, we show that the former is a generalized root stack of the latter. As an application, we define the period map for Gushel-Mukai varieties and construct some complete nonisotrivial families of smooth Gushel-Mukai varieties. In an appendix, we describe a generalization of the root stack construction used in our approach to the moduli space.

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arXiv2018

Derived categories of families of sextic del Pezzo surfaces

We construct a natural semiorthogonal decomposition for the derived category of an arbitrary flat family of sextic del Pezzo surfaces with at worst du Val singularities. This decomposition has three components equivalent to twisted derived categories of finite flat schemes of degrees 1, 3, and 2 over the base of the family. We provide a modular interpretation for these schemes and compute them explicitly in a number of standard families. For two such families the computation is based on a symmetric version of homological projective duality for $\mathbb{P}^2 \times \mathbb{P}^2$ and $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$, which we explain in an appendix.

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arXiv2018

System of Bibliometric Monitoring Sciences in Ukraine

The origins of scientometrics (research metrics) were analysed and the lack of attention to elaboration of its methodology was emphasized. The approaches to evaluation of scientific activity outcome were considered and the tendency of transition from formal quantitative indicators to receiving expert conclusion on the basis of bibliometric indicators was noted. The principles of the Leiden manifesto of scientometrics were set out, keeping to which provides clear monitoring and support of science development, and favours establishing of the constructive dialog between scientific environment and society as well. The conceptual statements and peculiarities of practical realization of the informative and analytical system "Bibliometryka Ukraynskoyi Nauky" ("Bibliometrics of the Ukrainian Science") elaborated by the Vernadsky National Library of Ukraine, are being represented. The proposals on the formation of advisory councils, which are aimed to adopt conclusions on the effectiveness of research activity of institutions, were considered. The feasibility of building a common platform for expert evaluation of scientific studies for countries of the Eastern Partnership by initiating similar bibliometric projects in these countries and their further convergence is proved.

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arXiv2018

Lower bound for the cost of connecting tree with given vertex degree sequence

The optimal connecting network problem generalizes many models of structure optimization known from the literature, including communication and transport network topology design, graph cut and graph clustering, structure identification from data, etc. For the case of connecting trees with the given sequence of vertex degrees, the cost of the optimal tree is shown to be bounded from below by the solution of a semidefinite optimization program with bilinear matrix constraints, which is reduced to the solution of a series of convex programs with linear matrix inequality constraints. The proposed lower bound estimate is used to construct several heuristic algorithms and to evaluate their quality on a variety of generated and real-life data sets. Keywords: Optimal communication network, generalized Wiener index, origin-destination matrix, semidefinite programming, quadratic matrix inequality.

math.CO↗
arXiv2018

Towards a cluster structure on trigonometric zastava

We study a moduli problem on a nodal curve of arithmetic genus 1, whose solution is an open subscheme in the zastava space for projective line. This moduli space is equipped with a natural Poisson structure, and we compute it in a natural coordinate system. We compare this Poisson structure with the trigonometric Poisson structure on the transversal slices in an affine flag variety. We conjecture that certain generalized minors give rise to a cluster structure on the trigonometric zastava.

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arXiv2018

On linear sections of the spinor tenfold, I

We discuss the geometry of transverse linear sections of the spinor tenfold $X$, the connected component of the orthogonal Grassmannian of 5-dimensional isotropic subspaces in a 10-dimensional vector space equipped with a non-degenerate quadratic form. In particular, we show that as soon as the dimension of a linear section of $X$ is at least 5, its integral Chow motive is of Lefschetz type. We discuss classification of smooth linear sections of $X$ of small codimension; in particular we check that there is a unique isomorphism class of smooth hyperplane sections and exactly two isomorphism classes of smooth linear sections of codimension 2. Using this, we define a natural quadratic line complex associated with a linear section of $X$. We also discuss the Hilbert schemes of linear spaces and quadrics on $X$ and its linear sections.

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arXiv2017

Hilbert schemes of lines and conics and automorphism groups of Fano threefolds

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 of index 1 and genus 10. We also show that the action of the automorphism group of a Fano threefold $X$ of index 2 (respectively, 1) on an irreducible component of its Hilbert scheme of lines (respectively, conics) is faithful if the anticanonical class of $X$ is very ample with a possible exception of several explicit cases. We use these faithfulness results to prove finiteness of the automorphism groups of most Fano threefolds and classify explicitly all Fano threefolds with infinite automorphism group. We also discuss a derived category point of view on the Hilbert schemes of lines and conics, and use this approach to identify some of them.

math.AG↗
arXiv2017

Derived categories of Gushel-Mukai varieties

We study the derived categories of coherent sheaves on Gushel-Mukai varieties. In the derived category of such a variety, we isolate a special semiorthogonal component, which is a K3 or Enriques category according to whether the dimension of the variety is even or odd. We analyze the basic properties of this category using Hochschild homology, Hochschild cohomology, and the Grothendieck group. We study the K3 category of a Gushel-Mukai fourfold in more detail. Namely, we show that this category is equivalent to the derived category of a K3 surface for a certain codimension 1 family of rational fourfolds, and to the K3 category of a birational cubic fourfold for a certain codimension 3 family. The first of these results verifies a special case of a duality conjecture which we formulate. We discuss our results in the context of the rationality problem for Gushel-Mukai varieties, which was one of the main motivations for this work.

math.AG↗
arXiv2017

Grothendieck ring of varieties, D- and L-equivalence, and families of quadrics

We discuss a conjecture saying that derived equivalence of simply connected smooth projective varieties implies that the difference of their classes in the Grothendieck ring of varieties is annihilated by a power of the affine line class. We support the conjecture with a number of known examples, and one new example. We consider a smooth complete intersection $X$ of three quadrics in ${\mathbf P}^5$ and the corresponding double cover $Y \to {\mathbf P}^2$ branched over a sextic curve. We show that as soon as the natural Brauer class on $Y$ vanishes, so that $X$ and $Y$ are derived equivalent, the difference $[X] - [Y]$ is annihilated by the affine line class.

math.AG↗
arXiv2017

Gushel-Mukai varieties: linear spaces and periods

Beauville and Donagi proved in 1985 that the primitive middle cohomology of a smooth complex cubic fourfold and the primitive second cohomology of its variety of lines, a smooth hyperkähler fourfold, are isomorphic as polarized integral Hodge structures. We prove analogous statements for smooth complex Gushel-Mukai varieties of dimension 4 (resp. 6), i.e., smooth dimensionally transverse intersections of the cone over the Grassmannian Gr(2,5), a quadric, and two hyperplanes (resp. of the cone over Gr(2,5) and a quadric). The associated hyperkähler fourfold is in both cases a smooth double cover of a hypersurface in ${\bf P}^5$ called an EPW sextic.

math.AG↗
arXiv2017

On quantum cohomology of Grassmannians of isotropic lines, unfoldings of $A_n$-singularities, and Lefschetz exceptional collections

The subject of this paper is the big quantum cohomology rings of symplectic isotropic Grassmannians $\text{IG}(2, 2n)$. We show that these rings are regular. In particular, by "generic smoothness", we obtain a conceptual proof of generic semisimplicity of the big quantum cohomology for $\text{IG}(2, 2n)$. Further, by a general result of Claus Hertling, the regularity of these rings implies that they have a description in terms of isolated hypersurface singularities, which we show in this case to be of type $A_{n-1}$. By the homological mirror symmetry conjecture, these results suggest the existence of a very special full exceptional collection in the derived category of coherent sheaves on $\text{IG}(2, 2n)$. Such a collection is constructed in the appendix by Alexander Kuznetsov.

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arXiv2017

Exceptional collections in surface-like categories

We provide a categorical framework for recent results of Markus Perling on combinatorics of exceptional collections on numerically rational surfaces. Using it we simplify and generalize some of Perling's results as well as Vial's criterion for existence of a numerical exceptional collection.

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arXiv2016

Gushel--Mukai varieties: classification and birationalities

We perform a systematic study of Gushel-Mukai varieties---quadratic sections of linear sections of cones over the Grassmannian Gr(2,5). This class of varieties includes Clifford general curves of genus 6, Brill-Noether general polarized K3 surfaces of genus 6, prime Fano threefolds of genus 6, and their higher-dimensional analogues. We establish an intrinsic characterization of normal Gushel-Mukai varieties in terms of their excess conormal sheaves, which leads to a new proof of the classification theorem of Gushel and Mukai. We give a description of isomorphism classes of Gushel-Mukai varieties and their automorphism groups in terms of linear algebraic data naturally associated to these varieties. We carefully develop the relation between Gushel-Mukai varieties and Eisenbud-Popescu-Walter sextics introduced earlier by Iliev-Manivel and O'Grady. We describe explicitly all Gushel-Mukai varieties whose associated EPW sextics are isomorphic or dual (we call them period partners or dual varieties respectively). Finally, we show that in dimension 3 and higher, period partners/dual varieties are always birationally isomorphic.

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