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We study Hilbert schemes of quadrics of dimension $k \in \{0,1,2,3\}$ on smooth Gushel-Mukai varieties $X$ of dimension $n \in \{2,3,4,5,6\}$ by relating them to the relative Hilbert schemes of linear subspaces of dimension $k + 1$ of a certain family, naturally associated with $X$, of quadrics of dimension $n - 1$ over the blowup of $\mathbf{P}^5$ at a point., Comment: 65 pages

We give a proof of Mukai's Theorem on the existence of certain exceptional vector bundles on prime Fano threefolds. To our knowledge this is the first complete proof in the literature. The result is essential for Mukai's biregular classification of prime Fano threefolds, and for the existence of semiorthogonal decompositions in their derived categories. Our approach is based on Lazarsfeld's construction that produces vector bundles on a variety from globally generated line bundles on a divisor, on Mukai's theory of stable vector bundles on K3 surfaces, and on Brill--Noether properties of curves and (in the sense of Mukai) of K3 surfaces., Comment: 38 pages

We classify all 1-nodal degenerations of smooth Fano threefolds with Picard number 1 (both nonfactorial and factorial) and describe their geometry. In particular, we describe a relation between such degenerations and smooth Fano threefolds of higher Picard rank and with unprojections of complete intersection varieties., Comment: 79 pages; v2: 1-cuspidal factorial Fano threefolds included into the classification; 83 pages

Using the technique of categorical absorption of singularities we prove that the nontrivial components of the derived categories of del Pezzo threefolds of degree $d \in \{2,3,4,5\}$ and crepant categorical resolutions of the nontrivial components of the derived categories of nodal del Pezzo threefolds of degree $d = 1$ can be smoothly deformed to the nontrivial components of the derived categories of prime Fano threefolds of genus $g = 2d + 2 \in \{4,6,8,10,12\}$. This corrects and proves the Fano threefolds conjecture of the first author from [Kuz09], and opens a way to interesting geometric applications, including a relation between the intermediate Jacobians and Hilbert schemes of curves of the above threefolds. We also describe a compactification of the moduli stack of prime Fano threefolds endowed with an appropriate exceptional bundle and its boundary component that corresponds to degenerations associated with del Pezzo threefolds., Comment: 39 pages; v4: final version

We give two simple geometric constructions of a smooth family of projective varieties with central fiber isomorphic to the horospherical variety of type $\mathrm{G}_2$ and all other fibers isomorphic to the isotropic orthogonal Grassmannian $\mathrm{OGr}(2,7)$ and discuss briefly the derived category of this family., Comment: 7 pages

In this paper we investigate homologically finite-dimensional objects in the derived category of a given small dg-enhanced triangulated category. Using these we define reflexivity, hfd-closedness, and the Gorenstein property for triangulated categories, and discuss crepant categorical contractions. We illustrate the introduced notions on examples of categories of geometric and algebraic origin and provide geometric applications. In particular, we apply our results to prove a bijection between semiorthogonal decompositions of the derived category of a singular variety and the derived category of its smoothing with support on the central fiber., Comment: Final version; accepted to Selecta

We introduce the notion of categorical absorption of singularities: an operation that removes from the derived category of a singular variety a small admissible subcategory responsible for singularity and leaves a smooth and proper category. We construct (under appropriate assumptions) a categorical absorption for a projective variety $X$ with isolated ordinary double points. We further show that for any smoothing $\mathcal{X}/B$ of $X$ over a smooth curve $B$, the smooth part of the derived category of $X$ extends to a smooth and proper over $B$ family of triangulated subcategories in the fibers of $\mathcal{X}$., Comment: 41 pages; v2, v3: minor improvements, v4: revised according to the referee's report, to appear in EPIGA, v5: published version

We study del Pezzo varieties, higher-dimensional analogues of del Pezzo surfaces. In particular, we introduce ADE classification of del Pezzo varieties, show that in type A the dimension of non-conical del Pezzo varieties is bounded by $12 - d - r$, where $d$ is the degree and $r$ is the rank of the class group, and classify maximal del Pezzo varieties., Comment: 62 pages, LaTeX, final version, to appear in Izvestiya

We construct $S$-linear semiorthogonal decompositions of derived categories of smooth Fano threefold fibrations $X/S$ with relative Picard rank $1$ and rational geometric fibers and discuss how the structure of components of these decompositions is related to rationality properties of $X/S$., Comment: 55 pages; v2: minor modifications

We discuss recent developments in the study of semiorthogonal decompositions of algebraic varieties with an emphasis on their behaviour in families. First, we overview new results concerning homological projective duality. Then we introduce residual categories, discuss their relation to small quantum cohomology, and compute Serre dimensions of residual categories of complete intersections. After that we define simultaneous resolutions of singularities and describe a construction that works in particular for nodal degenerations of even-dimensional varieties. Finally, we introduce the concept of absorption of singularities which works under appropriate assumptions for nodal degenerations of odd-dimensional varieties., Comment: To appear in the Proceedings of the ICM 2022; 41 pages; comments are welcome

We describe in terms of spherical twists the Serre functors of many interesting semiorthogonal components, called residual categories, of the derived categories of projective varieties. In particular, we show the residual categories of Fano complete intersections are fractional Calabi--Yau up to a power of an explicit spherical twist. As applications, we compute the Serre dimensions of residual categories of Fano complete intersections, thereby proving a corrected version of a conjecture of Katzarkov and Kontsevich, and deduce the nonexistence of Serre invariant stability conditions when the degrees of the complete intersection do not all coincide., Comment: 53 pages, minor updates

We introduce the notion of hyperbolic equivalence for quadric bundles and quadratic forms on vector bundles and show that hyperbolic equivalent quadric bundles share many important properties: they have the same Brauer data; moreover, if they have the same dimension over the base, they are birational over the base and have equal classes in the Grothendieck ring of varieties. Furthermore, when the base is a projective space we show that two quadratic forms are hyperbolic equivalent if and only if their cokernel sheaves are isomorphic up to twist, their fibers over a fixed point of the base are Witt equivalent, and, in some cases, certain quadratic forms on intermediate cohomology groups of the underlying vector bundles are Witt equivalent. For this we show that any quadratic form over $\mathbb{P}^n$ is hyperbolic equivalent to a quadratic form whose underlying vector bundle has many cohomology vanishings; this class of bundles, called VLC bundles in the paper, is interesting by itself., Comment: 42 pages; v2: minor improvements

We prove rationality criteria over algebraically non-closed fields of characteristic $0$ for five out of six types of geometrically rational Fano threefolds of Picard number $1$ and geometric Picard number bigger than $1$. For the last type of such threefolds we provide a unirationality criterion and prove stable non-rationality under additional assumptions., Comment: 36 pages, latex, final version, to appear in Journal of the Institute of Mathematics of Jussieu, a few typos are corrected

We introduce the notion of a simultaneous categorical resolution of singularities, a categorical version of simultaneous resolutions of rational double points of surface degenerations. Furthermore, we suggest a construction of simultaneous categorical resolutions which, in particular, applies to the case of a flat projective 1-dimensional family of varieties of arbitrarily high even dimension with ordinary double points in the total space and central fiber. As an ingredient of independent interest, we check that the property of a geometric triangulated category linear over a base to be relatively smooth and proper can be verified fiberwise. As an application we construct a smooth and proper family of K3 categories with general fiber the K3 category of a smooth cubic fourfold and special fiber the derived category of the K3 surface of degree 6 associated with a singular cubic fourfold., Comment: 23 pages; v4: a few typos corrected; to appear in Math. Z

For the derived category of the Cayley plane, which is the cominuscule Grassmannian of Dynkin type $\mathrm{E}_6$, 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 $\mathrm{F}_4$. We show that the restriction of the Faenzi-Manivel collection to such a hyperplane section gives a full Lefschetz exceptional collection, providing the first example of a full exceptional collection on a homogeneous variety of Dynkin type $\mathrm{F}$. We also describe the residual categories of these Lefschetz collections, confirming conjectures of the second and third named author for the Cayley plane and its hyperplane section. The latter description is based on a general result of independent interest, relating residual categories of a variety and its hyperplane section., Comment: Final version. To appear in Transformation Groups

We discuss birational properties of Mukai varieties, i.e., of higher-dimensional analogues of prime Fano threefolds of genus $g \in \{7,8,9,10\}$ over an arbitrary field $\mathsf{k}$ of zero characteristic. In the case of dimension $n \ge 4$ we prove that these varieties are $\mathsf{k}$-rational if and only if they have a $\mathsf{k}$-point except for the case of genus $9$, where we assume $n \ge 5$. Furthermore, we prove that Mukai varieties of genus $g \in \{7,8,9,10\}$ and dimension $n \ge 5$ contain cylinders if they have a $\mathsf{k}$-point. Finally, we prove that the embedding $X \hookrightarrow \mathrm{Gr}(3,7)$ for prime Fano threefolds of genus $12$ is defined canonically over any field and use this to give a new proof of the criterion of rationality., Comment: 34 pages, latex

We describe intermediate Jacobians of Gushel-Mukai varieties $X$ of dimensions 3 or 5: if $A$ is the Lagrangian space associated with $X$, we prove that the intermediate Jacobian of $X$ is isomorphic to the Albanese variety of the canonical double covering of any of the two dual Eisenbud-Popescu-Walter surfaces associated with $A$. As an application, we describe the period maps for Gushel-Mukai threefolds and fivefolds., Comment: 48 pages. Latest addition to our series of articles on the geometry of Gushel-Mukai varieties; v2: minor stylistic improvements, results unchanged; v3: minor improvements; v4: final version, published in EPIGA

In our previous paper we suggested a conjecture relating the structure of the small quantum cohomology ring of a smooth Fano variety to the structure of its derived category of coherent sheaves. Here we generalize this conjecture, make it more precise, and support by the examples of (co)adjoint homogeneous varieties of simple algebraic groups of Dynkin types $A_n$ and $D_n$, i.e., flag varieties $Fl(1,n;n+1)$ and isotropic orthogonal Grassmannians $OG(2,2n)$; in particular we construct on each of those an exceptional collection invariant with respect to the entire automorphism group. For $OG(2,2n)$ this is the first exceptional collection proved to be full., Comment: 31 pages; v2: introduction clarified; v3: final version

We give necessary and sufficient conditions for unirationality and rationality of Fano threefolds of geometric Picard rank-1 over an arbitrary field of zero characteristic., Comment: 62 pages, latex, final version, to appear in AJM

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., Comment: 48 pages. References updated.

We show that over an algebraically closed field of characteristic not equal to 2, homological projective duality for smooth quadric hypersurfaces and for double covers of projective spaces branched over smooth quadric hypersurfaces is a combination of two operations: one interchanges a quadric hypersurface with its classical projective dual and the other interchanges a quadric hypersurface with the double cover branched along it., Comment: 15 pages, minor updates

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., Comment: 51 pages

We develop an approach that allows to construct semiorthogonal decompositions of derived categories of surfaces with cyclic quotient singularities whose components are equivalent to derived categories of local finite dimensional algebras. We first explain how to induce a semiorthogonal decomposition of a surface $X$ with rational singularities from a semiorthogonal decomposition of its resolution. In the case when $X$ has cyclic quotient singularities, we introduce the condition of adherence for the components of the semiorthogonal decomposition of the resolution that allows to identify the components of the induced decomposition with derived categories of local finite dimensional algebras. Further, we present an obstruction in the Brauer group of $X$ to the existence of such semiorthogonal decomposition, and show that in the presence of the obstruction a suitable modification of the adherence condition gives a semiorthogonal decomposition of the twisted derived category of $X$. We illustrate the theory by exhibiting a semiorthogonal decomposition for the untwisted or twisted derived category of any normal projective toric surface depending on whether its Weil divisor class group is torsion-free or not. For weighted projective planes we compute the generators of the components explicitly and relate our results to the results of Kawamata based on iterated extensions of reflexive sheaves of rank 1., Comment: Exposition improved; Lemma 2.2 fixed

We introduce the notion of a categorical join, which can be thought of as a categorification of the classical join of two projective varieties. This notion is in the spirit of homological projective duality, which categorifies classical projective duality. Our main theorem says that the homological projective dual category of the categorical join is naturally equivalent to the categorical join of the homological projective dual categories. This categorifies the classical version of this assertion and has many applications, including a nonlinear version of the main theorem of homological projective duality., Comment: 58 pages. Final version, to appear in JAMS

We explain a general construction of double covers of quadratic degeneracy loci and Lagrangian intersection loci based on reflexive sheaves. We relate the double covers of quadratic degeneracy loci to the Stein factorizations of the relative Hilbert schemes of linear spaces of the corresponding quadric fibrations. We give a criterion for these double covers to be nonsingular. As applications of these results, we show that the double covers of the EPW sextics obtained by our construction give O'Grady's double EPW sextics and that an analogous construction gives Iliev-Kapustka-Kapustka-Ranestad's EPW cubes., Comment: 26 pages. As suggested by Nick Addington, we added an application to double covers of symmetroids. This third version also includes simple but useful additions in Theorem 5.2(2) and 5.7(2), and new Lemmas 3.3 and 4.3

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)., Comment: Final version. To appear in PLMS

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., Comment: 5 pages; v2: some details and references added

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., Comment: 43 pages; v2: exposition improved, some typos corrected, references and attributions included

We construct two small resolutions of singularities of the Coble fourfold (the double cover of the four-dimensional projective space branched over the Igusa quartic). We use them to show that all $S_6$-invariant three-dimensional quartics are birational to conic bundles over the quintic del Pezzo surface with the discriminant curves from the Wiman-Edge pencil. As an application, we check that $S_6$-invariant three-dimensional quartics are unirational, obtain new proofs of rationality of four special quartics among them and irrationality of the others, and describe their Weil divisor class groups as $S_6$-representations., Comment: 57 pages; v2: minor changes; v3: referee's comments taken into account; v4: published version

We give an explicit construction of prime Fano threefolds of genus 12 with a $G_m$-action, describe their isomorphism classes and automorphism groups., Comment: 14 pages, LaTeX, updated version, to appear in \'Epijournal de G\'eom\'etrie Alg\'ebrique, Vol. 2 (2018), Article Nr. 3

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., Comment: 48 pages; v2: minor improvements

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., Comment: Appendix by Alexander Kuznetsov. The paper incorporates the results of the earlier preprint arXiv:1510.07903. Comments are welcome

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., Comment: 23 pages; v2: some references added, the relation to the approach of [dTVdB] discussed

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., Comment: Exposition improved, main conjecture slightly updated

We prove that the derived category $D(C)$ of a generic curve of genus greater than one embeds into the derived category $D(M)$ of the moduli space $M$ of rank two stable bundles on $C$ with fixed determinant of odd degree., Comment: 20 pages; v2: exposition improved, some references added

We prove derived equivalence of Calabi-Yau threefolds constructed by Ito-Miura-Okawa-Ueda as an example of non-birational Calabi-Yau varieties whose difference in the Grothendieck ring of varieties is annihilated by the affine line., Comment: 5 pages

We provide a stable rationality construction for some smooth complex Gushel-Mukai varieties of dimension 6. As a consequence, we compute the integral singular cohomology of any smooth Gushel-Mukai sixfold and in particular, show that it is torsion-free., Comment: 13 pages

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., Comment: 43 pages, reorganized and edited

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 hyperk\"ahler 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 hyperk\"ahler fourfold is in both cases a smooth double cover of a hypersurface in ${\bf P}^5$ called an EPW sextic., Comment: 44 pages. Lemma 2.1 slightly expanded. Lemma 2.2, Lemma 3.3 (a Lefschetz-type result for cyclic coverings proved by Cornalba), and Lemma 3.7 are new. Other minor corrections

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., Comment: 60 pages, LaTeX, final version, prepared for publication, to appear in Japanese Journal of Mathematics, section 5 is essentially rewritten

We show that K\"uchle fivefolds of type (c5) --- subvarieties of the Grassmannian Gr(3,7) parameterizing 3-subspaces that are isotropic for a given 2-form and are annihilated by a given 4-form --- are birational to hyperplane sections of the Lagrangian Grassmannian LGr(3,6) and describe in detail these birational transformations. As an application, we show that the integral Chow motive of a K\"uchle fivefold of type (c5) is of Lefschetz type. We also discuss K\"uchle fourfolds of type (c5) --- hyperplane sections of the corresponding K\"uchle fivefolds --- an interesting class of Fano fourfolds, which is expected to be similar to the class of cubic fourfolds in many aspects., Comment: 34 pages

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., Comment: 61 pages. v2: Section 2 thoroughly rewritten. In particular, the definition of a GM variety become significantly more general. v3: minor inaccuracies corrected, minor changes in presentation

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., Comment: Lecture notes for the CIME-CIRM summer school, Levico Terme, June 22--27, 2015; 26 pages

We discuss Calabi-Yau and fractional Calabi-Yau semiorthogonal components of derived categories of coherent sheaves on smooth projective varieties. The main result is a general construction of a fractional Calabi-Yau category from a rectangular Lefschetz decomposition and a spherical functor. We give many examples of application of this construction and discuss some general properties of Calabi-Yau categories., Comment: 24 pages, v2: minor corrections

We study the natural Gieseker and Uhlenbeck compactifications of the rational Calogero-Moser phase space. The Gieseker compactification is smooth and provides a small resolution of the Uhlenbeck compactification. This allows computing the IC stalks of the Uhlenbeck compactification., Comment: 38 pages. v2: Remark 1.3.3 added. v3: The final version published in Selecta Mathematica

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., Comment: Main text by M. Finkelberg, A. Kuznetsov and L. Rybnikov with an appendix by G. Dobrovolska; v3 32 pages, Proof of Proposition 4.3 corrected, Section 1.5 added; v4 33 pages, the final version to appear in Selecta Math.; v5 the published version

We describe the geometry of K\"uchle varieties (i.e. Fano 4-folds of index 1 contained in the Grassmannians as zero loci of equivariant vector bundles) with Picard number greater than 1 and the structure of their derived categories., Comment: 10 pages, a reference added

Given a variety $Y$ with a rectangular Lefschetz decomposition of its derived category, we consider a degree $n$ cyclic cover $X \to Y$ ramified over a divisor $Z \subset Y$. We construct semiorthogonal decompositions of $\mathrm{D^b}(X)$ and $\mathrm{D^b}(Z)$ with distinguished components $\mathcal{A}_X$ and $\mathcal{A}_Z$, and prove the equivariant category of $\mathcal{A}_X$ (with respect to an action of the $n$-th roots of unity) admits a semiorthogonal decomposition into $n-1$ copies of $\mathcal{A}_Z$. As examples we consider quartic double solids, Gushel-Mukai varieties, and cyclic cubic hypersurfaces., Comment: 27 pages, minor changes

In this review we discuss what is known about semiorthogonal decompositions of derived categories of algebraic varieties. We review existing constructions, especially the homological projective duality approach, and discuss some related issues such as categorical resolutions of singularities., Comment: Contribution to the ICM 2014; v2: acknowledgements updated; v3: a reference added

A counterexample to the Jordan-H\"older property for semiorthogonal decompositions of derived categories of smooth projective varieties was constructed by B\"ohning, Graf von Bothmer and Sosna. In this short note we present a simpler example by realizing Bondal's quiver in the derived category of a blowup of the projective space.