2017
DOI: 10.1016/j.aim.2016.10.012
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Exceptional collections, and the Néron–Severi lattice for surfaces

Abstract: 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. … Show more

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Cited by 22 publications
(20 citation statements)
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“…Remark 10.12: Recent work by Vial [Via15] suggests that it should be possible to relax the conditions on our surface X by dropping the requirement that K 2 X = 12 − n. We will give a brief outline on how this condition can be dropped in many cases, though we are not able to remove it completely.…”
Section: 2mentioning
confidence: 99%
“…Remark 10.12: Recent work by Vial [Via15] suggests that it should be possible to relax the conditions on our surface X by dropping the requirement that K 2 X = 12 − n. We will give a brief outline on how this condition can be dropped in many cases, though we are not able to remove it completely.…”
Section: 2mentioning
confidence: 99%
“…This question, which was the original motivation for the current work, is now central to a growing body of research into arithmetic aspects of the theory of derived categories, see . As an example, if S is a smooth projective surface, Hassett and Tschinkel [, Lemma 8] prove that the index of S can be recovered from sans-serifDnormalbfalse(Sfalse) as the greatest common divisor of the second Chern classes of objects.…”
Section: Introductionmentioning
confidence: 99%
“…For del Pezzo surfaces, this lattice is a twist of a semisimple root lattice with the Galois action factoring through the Weyl group, see [, Theorem 2.12]. Vial has recently studied how one can, given a k‐exceptional collection on a surface S, obtain information on the Néron–Severi lattice. In particular, he shows that a geometrically rational surface with an exceptional collection of maximal length is rational.…”
Section: Introductionmentioning
confidence: 99%
“…We complete our list of generators of Pic X g by introducing G g 10 . The choice of G g 10 is motivated by the proof of the step (iii) ⇒ (i) in [34,Theorem 3.1].…”
Section: The Néron-severi Lattices Of Dolgachev Surfaces Of Type (2 3)mentioning
confidence: 99%