Full Exceptional Collection Research Articles

Mutations of numerically exceptional collections on surfaces

A conjecture of Bondal–Polishchuk states that, in particular for the bounded derived category of coherent sheaves on a smooth projective variety, the action of the braid group on full exceptional collections is transitive up to shifts. We show that the braid group acts transitively on the set of maximal numerically exceptional collections on rational surfaces up to isometries of the Picard lattice and twists with line bundles. Considering the blow-up of the projective plane in up to 9 points in very general position, these results lift to the derived category. More precisely, we prove that, under these assumptions, a maximal numerically exceptional collection consisting of line bundles is a full exceptional collection and any two of them are related by a sequence of mutations and shifts. The former extends a result of Elagin–Lunts and the latter a result of Kuleshov–Orlov, both concerning del Pezzo surfaces. In contrast, we show in concomitant work (Krah in Invent Math 235(3):1009–1018, 2024) that the blow-up of the projective plane in 10 points in general position admits a non-full exceptional collection of maximal length consisting of line bundles.

On the derived category of the Cayley Grassmannian

We construct a full exceptional collection consisting of vector bundles in the derived category of coherent sheaves on the so-called Cayley Grassmannian, the subvariety of the Grassmannian Gr(3,7) parameterizing 3-subspaces that are annihilated by a general 4-form. The main step in the proof of fullness is a construction of two self-dual vector bundles which is obtained from two operations with quadric bundles that might be interesting in themselves.

Exceptional collections for mirrors of invertible polynomials

We prove the existence of a full exceptional collection for the derived category of equivariant matrix factorizations of an invertible polynomial with its maximal symmetry group. This proves a conjecture of Hirano–Ouchi. In the Gorenstein case, we also prove a stronger version of this conjecture due to Takahashi. Namely, that the full exceptional collection is strong.

Admissible subcategories of del Pezzo surfaces

We study admissible subcategories of derived categories of coherent sheaves on del Pezzo surfaces and rational elliptic surfaces. Using a relation between admissible subcategories and anticanonical divisors we prove the following results. First, we classify all admissible subcategories of the projective plane by showing that each is generated by a subcollection of a full exceptional collection. Second, we show that the derived categories of del Pezzo surfaces do not contain any phantom subcategories. This provides first examples of varieties of dimension larger than one that have some nontrivial admissible subcategories, but provably do not contain phantoms. We also prove that any admissible subcategory supported set-theoretically on a smooth (−1)-curve in a surface is generated by some twist of the structure sheaf of that curve.

Gamma conjecture II for quadrics

The Gamma conjecture II for the quantum cohomology of a Fano manifold F, proposed by Galkin, Golyshev and Iritani, describes the asymptotic behavior of the flat sections of the Dubrovin connection near the irregular singularities, in terms of a full exceptional collection, if there exists, of \({\mathcal {D}}^b(F)\) and the \({\widehat{\Gamma }}\)-integral structure. In this paper, for the smooth quadric hypersurfaces we prove the convergence of the full quantum cohomology and Gamma II. For the proof, we first give a criterion on Gamma II for Fano manifolds with semisimple quantum cohomology, by Dubrovin’s theorem of analytic continuations of semisimple Frobenius manifolds. Then we work out a closed formula of the Chern characters of spinor bundles on quadrics. By the deformation-invariance of Gromov–Witten invariants we show that the full quantum cohomology can be reconstructed by its ambient part, and use this to obtain estimations. Finally we complete the proof of Gamma II for quadrics by explicit asymptotic expansions of flat sections corresponding to Kapranov’s exceptional collections and an application of our criterion.

Proper connective differential graded algebras and their geometric realizations

We show that every proper connective dg algebra A admits a geometric realization (as defined by Orlov) by a smooth projective scheme with a full exceptional collection. If A is moreover smooth, we compute the noncommutative Chow motive of A. We go on to analyse the relationship between smoothness and regularity in more detail as well as commenting on smoothness of the degree zero cohomology for smooth proper connective dg algebras.

No phantoms in the derived category of curves over arbitrary fields, and derived characterizations of Brauer–Severi varieties

We show that the derived category of Brauer–Severi curves satisfies the Jordan–Holder property and cannot have quasi-phantoms, phantoms or universal phantoms. In this way we obtain that quasi-phantoms, phantoms or universal phantoms cannot exist in the derived category of smooth projective curves over a field k. Moreover, we show that a n-dimensional Brauer–Severi variety is completely characterized by the existence of a full weak exceptional collection consisting of pure vector bundles of length n+1, at least in characteristic zero. We conjecture that Brauer–Severi varieties X satisfy rdimcat(X)= ind(X)−1, provided period equals index, and prove this in the case of curves, surfaces and for Brauer–Severi varieties of index at most three. We believe that the results for curves are known to the experts. We nevertheless give the proofs, adding to the literature.

DERIVED CATEGORIES OF THE CAYLEY PLANE AND THE COADJOINT GRASSMANNIAN OF TYPE F

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 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 F.

Lefschetz exceptional collections in $$S_k$$-equivariant categories of $$(\mathbb {P}^n)^{k}$$

We consider the bounded derived category of \(S_k\)-equivariant coherent sheaves on \((\mathbb {P}^n)^k\). The goal of this paper is to construct in this category a rectangular Lefschetz exceptional collection of maximal length when this is possible, or a minimal Lefschetz exceptional collection when a rectangular one does not exist. The main results of the paper include the construction of a full rectangular Lefschetz exceptional collection in the case \(k=3\) and in the case \(n=1\) when \(\mathrm {gcd}(n+1,k)=1\). We also construct a full minimal Lefschetz exceptional collection for \(n=1\) and even k, and for \(n=2\) and \(k=3\).

Geometry of Horospherical Varieties of Picard Rank One

AbstractWe study the geometry of smooth non-homogeneous horospherical varieties of Picard rank one. These have been classified by Pasquier and include the well-known odd symplectic Grassmannians. We focus our study on quantum cohomology, with a view towards Dubrovin’s conjecture. We start with describing the cohomology groups of smooth horospherical varieties of Picard rank one. We show a Chevalley formula for these and establish that many Gromov–Witten invariants are enumerative. This enables us to prove that in many cases the quantum cohomology is semisimple. We give a presentation of the quantum cohomology ring for odd symplectic Grassmannians. In the last sections, we turn to derived categories of coherent sheaves. We first discuss a general construction of exceptional bundles on horospherical varieties. We work out in detail the case of the horospherical variety associated to the exceptional group $G_2$ and construct a full rectangular Lefschetz exceptional collection in the derived category.

Kernels for Grassmann flops

We develop a generalization of the Q-construction of the first author, Diemer, and the third author for Grassmann flips. This generalization provides a canonical idempotent kernel on the derived category of the associated global quotient stack. The idempotent kernel, after restriction, induces a semi-orthogonal decomposition which compares the flipped varieties. Furthermore its image, after restriction to the geometric invariant theory semistable locus, “opens” a canonical “window” in the derived category of the quotient stack. We check this window coincides with the set of representations used by Kapranov to form a full exceptional collection on Grassmannians.

Exceptional collections on certain Hassett spaces

We construct an $S_2\times S_n$ invariant full exceptional collection on Hassett spaces of weighted stable rational curves with $n+2$ markings and weights $(\frac{1}{2}+\eta, \frac{1}{2}+\eta,\epsilon,\ldots,\epsilon)$, for $0<\epsilon, \eta\ll1$ and can be identified with symmetric GIT quotients of $(\mathbb{P}^1)^n$ by the diagonal action of $\mathbb{G}_m$ when $n$ is odd, and their Kirwan desingularization when $n$ is even. The existence of such an exceptional collection is one of the needed ingredients in order to prove the existence of a full $S_n$-invariant exceptional collection on $\overline{\mathcal{M}}_{0,n}$. To prove exceptionality we use the method of windows in derived categories. To prove fullness we use previous work on the existence of invariant full exceptional collections on Losev-Manin spaces.

Derived category of moduli of pointed curves. I

This is the first paper in the sequence devoted to derived category of moduli spaces of curves of genus $0$ with marked points. We develop several approaches to describe it equivariantly with respect to the action of the symmetric group permuting marked points. We construct an equivariant full exceptional collection on the Losev-Manin space which categorifies derangements.

On cyclic strong exceptional collections of line bundles on surfaces

We study exceptional collections of line bundles on surfaces. We prove that any full cyclic strong exceptional collection of line bundles on a rational surface is an augmentation in the sense of Lutz Hille and Markus Perling. We find simple geometric criteria of exceptionality (strong exceptionality, cyclic strong exceptionality) for collections of line bundles on weak del Pezzo surfaces. As a result, we classify smooth projective surfaces admitting a full cyclic strong exceptional collection of line bundles. Also, we provide an example of a weak del Pezzo surface of degree 2 and a full strong exceptional collection of line bundles on it which does not come from augmentations, thus answering a question by Hille and Perling.

ON THE BOUNDED DERIVED CATEGORY OF IGr(3, 7)

We construct a mininal Lefschetz decomposition of the bounded derived category of coherent sheaves on the isotropic Grassmannian $\mathsf{IGr}(3,7)$. Moreover, we show that $\mathsf{IGr}(3, 7)$ admits a full exceptional collection consisting of equivariant vector bundles.

Maximally-graded matrix factorizations for an invertible polynomial of chain type

In 1977, Orlik–Randell construct a nice integral basis of the middle homology group of the Milnor fiber associated to an invertible polynomial of chain type and they conjectured that it is represented by a distinguished basis of vanishing cycles.The purpose of this paper is to prove the algebraic counterpart of the Orlik–Randell conjecture. Under the homological mirror symmetry, we may expect that the triangulated category of maximally-graded matrix factorizations for the Berglund–Hübsch transposed polynomial admits a full exceptional collection with a nice numerical property. Indeed, we show that the category admits a Lefschetz decomposition with respect to a polarization in the sense of Kuznetsov–Smirnov, whose Euler matrix are calculated in terms of the “zeta function” of the inverse of the polarization.As a corollary, it turns out that the homological mirror symmetry holds at the level of the Grothendieck group in the following sense; the Grothendieck group of the category with the Euler form is isomorphic to the middle homology group with the (quasi-)Seifert form (with a suitable sign).

On the derived category of

We construct a full exceptional collection of vector bundles in the bounded derived category of coherent sheaves on the Grassmannian of isotropic 3-dimensional subspaces in a symplectic vector space of dimension 8. Bibliography: 16 titles.

A maximally-graded invertible cubic threefold that does not admit a full exceptional collection of line bundles

Abstract We show that there exists a cubic threefold defined by an invertible polynomial that, when quotiented by the maximal diagonal symmetry group, has a derived category that does not have a full exceptional collection consisting of line bundles. This provides a counterexample to a conjecture of Lekili and Ueda.

TILTING THEORY FOR GORENSTEIN RINGS IN DIMENSION ONE

In representation theory, commutative algebra and algebraic geometry, it is an important problem to understand when the triangulated category$\mathsf{D}_{\operatorname{sg}}^{\mathbb{Z}}(R)=\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R$admits a tilting (respectively, silting) object for a$\mathbb{Z}$-graded commutative Gorenstein ring$R=\bigoplus _{i\geqslant 0}R_{i}$. Here$\mathsf{D}_{\operatorname{sg}}^{\mathbb{Z}}(R)$is the singularity category, and$\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R$is the stable category of$\mathbb{Z}$-graded Cohen–Macaulay (CM)$R$-modules, which are locally free at all nonmaximal prime ideals of$R$.In this paper, we give a complete answer to this problem in the case where$\dim R=1$and$R_{0}$is a field. We prove that$\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R$always admits a silting object, and that$\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R$admits a tilting object if and only if either$R$is regular or the$a$-invariant of$R$is nonnegative. Our silting/tilting object will be given explicitly. We also show that if$R$is reduced and nonregular, then its$a$-invariant is nonnegative and the above tilting object gives a full strong exceptional collection in$\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R=\text{}\underline{\mathsf{CM}}^{\mathbb{Z}}R$.

Classification of full exceptional collections of line bundles on three blow-ups of $\mathbb{P}^{3}$

A fullness conjecture of Kuznetsov says that if a smooth pro-jective variety X admits a full exceptional collection of line bundles oflength l, then any exceptional collection of line bundles of le ...