Exceptional collections on Dolgachev surfaces associated with degenerations

Cho, Yonghwa; Lee, Yongnam

doi:10.48550/arxiv.1506.05213

Cited by 7 publications

(12 citation statements)

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“…N.B. Cho and Lee [10] have recently constructed exceptional collections on some Dolgachev surfaces of type X 9 (2, 3) of maximal length whose orthogonal complements provide examples of phantom categories.…”

Section: Numerically Exceptional Collections Of Maximal Length For Co...mentioning

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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Section: Numerically Exceptional Collections Of Maximal Length For Co...mentioning

confidence: 99%

Exceptional collections, and the Néron–Severi lattice for surfaces

Vial

2017

Advances in Mathematics

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“…Then A is a universal phantom category. Similarly, if X 9 (2, 3) is the Dolgachev surface considered in [8], then the phantom it admits is universal. Furthermore, the derived categories of the Hilbert schemes S [n] and (X 9 (2, 3)) [n] of n points on these surfaces admit a universal phantom.…”

Section: On Phantoms and Motivesmentioning

confidence: 99%

“…With regards to Question 3: most examples so far were constructed in Kodaira dimension 2. In the recent preprint [8] the authors construct a phantom on a Dolgachev surface which has Kodaira dimension 1. Since K3 and abelian surfaces are Calabi-Yau, they do not admit any semi-orthogonal decompositions (this statement uses [14,Prop.…”

Section: Possible Further Directionsmentioning

confidence: 99%

“…Recently, very special examples of semi-orthogonal decompositions were constructed. Namely, it was shown for several complex surfaces X that D b (X) admits a semi-orthogonal decomposition consisting of an exceptional collection of line bundles whose orthogonal complement is a category with trivial Hochschild homology and finite or trivial Grothendieck group; see, for example, [1], [3], [4], [8], [11], [12] or [22]. In the first case we call such a category a quasi-phantom and in the second case a phantom.…”

Section: Introductionmentioning

confidence: 99%

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Some remarks on phantom categories and motives

Sosna¹

2015

Preprint

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“…Quasiphantom categories are surprising new subcategories in the derived categories of algebraic varieties first discovered by Böhning, Bothmer and Sosna in [7]. Their discovery provides new perspectives on the study of derived categories of algebraic varieties and recently many examples of quasiphantom categories were constructed by many authors(see [1,6,7,10,11,15,16,18,19,20,22,23,24] for more details). However their structures are quite mysterious and we do not know whether every surface of general type with p g = q = 0 has a quasiphantom category in its derived category.…”

Section: Introductionmentioning

confidence: 98%

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

Kim

Lee

2017

Journal of Algebra

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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 p g = 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.A surface S which is isomorphic to (C ×D)/G where C, D are curves of genus ≥ 2 and G is a finite group acting on C × D freely is called a surface isogenous to a higher product. Surfaces isogenous to a higher product are interesting and important classes of surfaces of general type. They play an important role in the study of moduli spaces of general type surfaces. Bauer, Catanese and Grunewald classified surfaces isogenous to a higher product of unmixed type with p g = q = 0 in [4]. In particular, they proved that the possible list ofA 5 cases. It is very natural to expect that derived categories of all surfaces isogenous to a higher product with p g = q = 0 have quasiphantom categories in their derived categories. However quasiphantom categories in derived categories of surfaces had not been constructed for G = G(32, 27), A 5 cases.In this paper 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 p g = q = 0 and G is G(32, 27). From these exceptional collections we can obtain new examples of quasiphantom categories. They are obtained as the orthogonal complements of the exceptional collections of maximal length 4 on S.Acknowledgement. We are grateful to Changho Keem, Young-Hoon Kiem for their words of encouragement. We thank Fabrizio Catanese for answering many questions and helpful discussions. We also thank Alexander Kuznetsov for his suggestion to use a computer program to compute the derived categories of the family of surfaces considered in this paper. Part of this work was done when the third named author was a research fellow of Korea Institute for Advanced Study. He thanks KIAS for wonderful working condition and kind hospitality.This work was also supported by IBS-R003-Y1. Last but not least, we thank the referees for their valuable comments which helped us to improve the manuscript.Notations. We will work over C. Derived category of a variety will mean the

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Exceptional collections on Dolgachev surfaces associated with degenerations

Cited by 7 publications

References 24 publications

Exceptional collections, and the Néron–Severi lattice for surfaces

Exceptional collections, and the Néron–Severi lattice for surfaces

Some remarks on phantom categories and motives

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

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