Exceptional Collection Of Line Bundles Research Articles

On strong exceptional collections of line bundles of maximal length on fano toric Deligne–Mumford stacks

We study strong exceptional collections of line bundles on Fano toric Deligne-Mumford stacks $\mathbb{P}_{\mathbf{\Sigma}}$ with rank of Picard group at most two. We prove that any strong exceptional collection of line bundles generates the derived category of $\mathbb{P}_{\mathbf{\Sigma}}$, as long as the number of elements in the collection equals the rank of the (Grothendieck) $K$-theory group of $\mathbb{P}_{\mathbf{\Sigma}}$.

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 Landau–Ginzburg systems, co-tropical geometry, and Db(X) of various toric Fano manifolds

Let X be a toric Fano manifold given by a reflexive polytope Δ and let Crit(f)⊂(C*)s be the solution scheme of the Landau–Ginzburg system associated with a Laurent polynomial fu(z)≔∑n∈Δ◦(0)∩Zsunzn∈L(Δ◦). Motivated by mirror symmetry, we construct a map E : Crit(fu) → Pic(X) and show various cases of (X, fu) for which its image Eu(X)≔E(z)|z∈Crit(fu)⊂Pic(X) is a full strongly exceptional collection of line bundles. Moreover, we study relations between Hom(E(z), E(w)) for z, w ∈ Crit(fu) and the structure of the monodromy group acting on Crit(fu). The construction relies on a study of the tropical and co-tropical properties of Crit(fu) as log|u| → ±∞.

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.

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 ...

Applications of toric systems on surfaces

In this note we apply the techniques of the toric systems introduced by Hille–Perling to several problems on smooth projective surfaces: We showed that the existence of full exceptional collection of line bundles implies the rationality for small Picard rank surfaces; we proved equivalences of several notions of cyclic strong exceptional collection of line bundles; we also proposed a partial solution to a conjecture on exceptional sheaves on weak del Pezzo surfaces.

The McKay Correspondence, Tilting, and Rationality

We consider the problem of comparing t-structures under the derived McKay correspondence and for tilting equivalences. In low-dimensional cases, we relate the t-structures via torsion theories arising from additive functions on the triangulated category. As an application, we give a criterion for rationality for surfaces with a tilting bundle. We also show that every smooth projective surface that admits a full, strong, and exceptional collection of line bundles is rational.

On exceptional collections of line bundles and mirror symmetry for toric Del-Pezzo surfaces

Let X be a toric Del-Pezzo surface and let Crit(W)⊂(ℂ*)n be the solution scheme of the Landau-Ginzburg system of equations. Denote by X° the polar variety of X. Our aim in this work is to describe a map L:Crit(W)→Fuktrop(X°) whose image under homological mirror symmetry corresponds to a full strongly exceptional collection of line bundles.

Differential Graded Quivers of Smooth Rational Surfaces

AbstractLetXbe a smooth rational surface. We calculate a differential graded (DG) quiver of a full exceptional collection of line bundles onXobtained by an augmentation from a strong exceptional collection on the minimal model ofX. In particular, we calculate canonical DG algebras of smooth toric surfaces.

Quasiphantom categories on a family of surfaces isogenous to a higher product

We construct exceptional collections of line bundles of maximal length 4 on S=(C×D)/G which is a surface isogenous to a higher product with pg=q=0 where G=G(32,27) is a finite group of order 32 having number 27 in the list of Magma library. From these exceptional collections, we obtain new examples of quasiphantom categories as their orthogonal complements.

Tilting bundles on toric Fano fourfolds

This paper constructs tilting bundles obtained from full strong exceptional collections of line bundles on all smooth 4-dimensional toric Fano varieties. The tilting bundles lead to a large class of explicit Calabi–Yau-5 algebras, obtained as the corresponding rolled-up helix algebra. A database of the full strong exceptional collections can be found in the package QuiversToricVarieties for the computer algebra system Macaulay2.

On Landau–Ginzburg systems, quivers and monodromy

Let X be a toric Fano manifold and denote by C r i t ( f X ) ⊂ ( C ∗ ) n the solution scheme of the corresponding Landau–Ginzburg system of equations. For toric Del-Pezzo surfaces and various toric Fano threefolds we define a map L : C r i t ( f X ) → P i c ( X ) such that E L ( X ) : = L ( C r i t ( f X ) ) ⊂ P i c ( X ) is a full strongly exceptional collection of line bundles. We observe the existence of a natural monodromy map M : π 1 ( L ( X ) ∖ R X , f X ) → A u t ( C r i t ( f X ) ) where L ( X ) is the space of all Laurent polynomials whose Newton polytope is equal to the Newton polytope of f X , the Landau–Ginzburg potential of X , and R X ⊂ L ( X ) is the space of all elements whose corresponding solution scheme is reduced. We show that monodromies of C r i t ( f X ) admit non-trivial relations to quiver representations of the exceptional collection E L ( X ) . We refer to this property as the M -aligned property of the maps L : C r i t ( f X ) → P i c ( X ) . We discuss possible applications of the existence of such M -aligned exceptional maps to various aspects of mirror symmetry of toric Fano manifolds.

Enumerating exceptional collections of line bundles on some surfaces of general type

We use constructions of surfaces as abelian covers to write down exceptional collections of line bundles of maximal length for every surface X in certain families of surfaces of general type with pg = 0 and K 2 = 3,4,5,6,8. We also compute the algebra of derived endomorphisms for an appropriately chosen exceptional collection, and the Hochschild cohomology of the corresponding quasiphantom category. As a consequence, we see that the subcategory generated by the exceptional collection does not vary in the family of surfaces. Fi- nally, we describe the semigroup of effective divisors on each surface, answering a question of Alexeev.

Maximal lengths of exceptional collections of line bundles

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 $c>\frac34$ there exist infinitely many toric Fano varieties $Y$ with Picard number three, such that the maximal length of exceptional collection of line bundles on $Y$ is strictly less than $c\rk K_0(Y).$ To obtain varieties without exceptional collections of line bundles, it suffices to put $c=1.$ On the other hand, we prove that for any toric nef-Fano DM stack $Y$ with Picard number three, there exists a strong exceptional collection of line bundles on $Y$ of length at least $\frac34 \rk K_0(Y).$ The constant $\frac34$ is thus maximal with this property.

Exceptional collections of line bundles on the Beauville surface

We construct quasi-phantom admissible subcategories in the derived category of coherent sheaves on the Beauville surface S. These quasi-phantoms subcategories appear as right orthogonals to subcategories generated by exceptional collections of maximal possible length 4 on S. We prove that there are exactly 6 exceptional collections consisting of line bundles (up to a twist) and these collections are spires of two helices.

Exceptional collections of line bundles on projective homogeneous varieties

We construct new examples of exceptional collections of line bundles on the variety of Borel subgroups of a split semisimple linear algebraic group G of rank 2 over a field. We exhibit exceptional collections of the expected length for types A2 and B2=C2 and prove that no such collection exists for type G2. This settles the question of the existence of full exceptional collections of line bundles on projective homogeneous G-varieties for split linear algebraic groups G of rank at most 2.

Mori Dream Spaces as fine moduli of quiver representations

We construct Mori Dream Spaces as fine moduli spaces of representations of bound quivers, thereby extending results of Craw–Smith [6] beyond the toric case. Any collection of effective line bundles ℒ=(풪X,L1,…,Lr) on a Mori Dream Space X defines a bound quiver of sections and a map from X to a toric quiver variety |ℒ| called the multigraded linear series. We provide necessary and sufficient conditions for this map to be a closed immersion and, under additional assumptions on ℒ, the image realises X as the fine moduli space of ϑ-stable representations of the bound quiver. As an application, we show how to reconstruct del Pezzo surfaces from a full, strongly exceptional collection of line bundles.

Derived category of toric varieties with small Picard number

This paper aims to construct a full strongly exceptional collection of line bundles in the derived category D b (X), where X is the blow up of ℙ n−r ×ℙ r along a multilinear subspace ℙ n−r−1×ℙ r−1 of codimension 2 of ℙ n−r ×ℙ r . As a main tool we use the splitting of the Frobenius direct image of line bundles on toric varieties.

On the conjecture of King for smooth toric Deligne–Mumford stacks

We construct full strong exceptional collections of line bundles on smooth toric Fano Deligne–Mumford stacks of Picard number at most two and of any Picard number in dimension two. It is hoped that the approach of this paper will eventually lead to the proof of the existence of such collections on all smooth toric nef-Fano Deligne–Mumford stacks.

Brane tilings and exceptional collections

Both brane tilings and exceptional collections are useful tools for describing the low energy gauge theory on a stack of D3-branes probing a Calabi-Yau singularity. We provide a dictionary that translates between these two heretofore unconnected languages. Given a brane tiling, we compute an exceptional collection of line bundles associated to the base of the non-compact Calabi-Yau threefold. Given an exceptional collection, we derive the periodic quiver of the gauge theory which is the graph theoretic dual of the brane tiling. Our results give new insight to the construction of quiver theories and their relation to geometry.