Exceptional collections of line bundles on the Beauville surface

Galkin, Sergey; Shinder, Evgeny

doi:10.1016/j.aim.2013.06.007

Cited by 44 publications

(58 citation statements)

References 11 publications

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“…We think that we can generalize this result to any surface isogenous to a higher product with p g = q = 0. The following conjecture has also appeared in [12]. Conjecture 1.2.…”

Section: Introductionmentioning

confidence: 93%

“…Therefore the maximal possible length of the exceptional sequence on every such surface is 4. For the 4 families of such surfaces with abelian group quotients, exceptional collections of maximal length were constructed in [12,15,16]. In this paper we construct such collections in four more cases where G is D 4 ×Z/2, S 4 , S 4 ×Z/2 and (Z/4×Z/2)⋊Z/2 (G(16) in the notation of [6]).…”

Section: Introductionmentioning

confidence: 99%

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Exceptional collections on some fake quadrics

Lee

Shabalin

2018

Proc. Amer. Math. Soc.

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We construct exceptional collections of maximal length on four families of surfaces of general type with pg = q = 0 which are isogenous to a product of curves. From these constructions we obtain new examples of quasiphantom categories as their orthogonal complements.Question. Let S be a smooth projective surface of general type with p g = q = 0. Is there an exceptional sequence whose length is equal to the rank of Grothendieck Key words and phrases. Derived category, exceptional sequence, quasiphantom category, surfaces of general type, surfaces isogenous to a higher product.

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“…We think that we can generalize this result to any surface isogenous to a higher product with p g = q = 0. The following conjecture has also appeared in [12]. Conjecture 1.2.…”

Section: Introductionmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

Exceptional collections on some fake quadrics

Lee

Shabalin

2018

Proc. Amer. Math. Soc.

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“…An immediate corollary of this theorem is that a surface admitting a cyclic strong exceptional collection of line bundles of maximal length must be a rational surface. An alternating proof of this fact is given in [12,Lemma 15.1] Motivated by the desire to describe geometric phantom categories: subcategories of D b (X) with trivial Grothendieck group K 0 and Hochschild homology HH 0 , a large amount of work was carried out to establish the exceptional collection of line bundles of maximal length on a surface of general type with p g = q = 0, see [1], [4], [13], [14]. It immediately follows from Theorem 1.2 that: Corollary 1.3.…”

Section: Introductionmentioning

confidence: 99%

Applications of toric systems on surfaces

Zhang

2019

Journal of Pure and Applied Algebra

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“…Numerical constraints. Although the problem of classifying smooth projective complex surfaces that admit a full exceptional collection seems out of reach at present (it is conjectured that only the surfaces that are rational have a full exceptional collection), a fair amount of work [1,7,8,16] has been carried out in order to construct exceptional collections of maximal length on complex surfaces with p g = q = 0. (As usual, for a smooth projective surface S, the geometric genus is p g := h 0 (Ω 2 S ) = h 2 (O S ) and the irregularity is q := h 1 (O S ).)…”

Section: Introductionmentioning

confidence: 99%

Exceptional collections, and the Néron–Severi lattice for surfaces

Vial

2017

Advances in Mathematics

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We work out properties of smooth projective varieties X over a (not necessarily algebraically closed) field k that admit collections of objects in the bounded derived category of coherent sheaves D b (X) that are either full exceptional, or numerically exceptional of maximal length. Our main result gives a necessary and sufficient condition on the Néron-Severi lattice for a smooth projective surface S with χ(O S ) = 1 to admit a numerically exceptional collection of maximal length, consisting of line-bundles. As a consequence we determine exactly which complex surfaces with pg = q = 0 admit a numerically exceptional collection of maximal length. Another consequence is that a minimal geometrically rational surface with a numerically exceptional collection of maximal length is rational.As is apparent, the class of varieties admitting full exceptional collections is rather restricted. Perhaps the simplest constraint for a smooth projective variety to admit a full exceptional collection is the following. If we have a semi-orthogonal decomposition T = A, B , then. , E r is a full exceptional collection, then K 0 (X) = Z r . As will be explained in the introduction of Section 2, this implies that the Chow motive of X with rational coefficients is a direct sum of Lefschetz motives. Such a constraint was originally obtained via the theory of non-commutative motives by Marcolli and Tabuada [34].Chow motives and Lefschetz motives. Let R be a ring. The category of Chow motives with R-coefficients over k is constructed as follows. First one linearizes the category of smooth projective varieties over k by declaring that Hom(X, Y ) = CH d (X × Y ) ⊗ Z R, that is, by declaring that the morphism between X and Y are given by correspondences with R-coefficients modulo rational equivalence. Here, X is assumed to be of pure dimension d (otherwise one works component-wise) and the composition law is given byHere, e is the dimension of Y , and p XZ , p XY , p Y Z are the projections from X × Y × Z onto X × Z, X × Y, Y × Z, respectively. This R-linear category is far from being abelian, so that one formally adds to this R-linear category the images of idempotents. This is called taking the pseudo-abelian, or Karoubi, envelope. This new category is called the category of effective Chow motives, and objects are pairs (X, p), where X is a smooth projective variety of dimension d and p ∈ CH d (X × X) ⊗ Z R is an idempotent. When p is the class of the diagonal ∆ X in CH d (X × X), we write h(X) for (X, ∆ X ). In general, the object (X, p) should be thought of as the image of p acting on the motive h(X) of X. For example, in the category of effective Chow motives, the motive h(P 1 ) of the projective line becomes isomorphic to (P 1 , p) ⊕ (P 1 , q), where p := {0} × P 1 and q := P 1 × {0} are idempotents in Hom(h(P 1 ), h(P 1 )) := CH 1 (P 1 × P 1 ). The object (P 1 , p) is isomorphic to 1 := h(Spec k), and the motive (P 1 , p) is called the Lefschetz motive and is written 1(−1). The fiber product of two smooth projective varieties induces a...

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Exceptional collections of line bundles on the Beauville surface

Cited by 44 publications

References 11 publications

Exceptional collections on some fake quadrics

Exceptional collections on some fake quadrics

Applications of toric systems on surfaces

Exceptional collections, and the Néron–Severi lattice for surfaces

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