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. 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...

Combinatorial aspects of exceptional sequences on (rational) surfaces

“…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.…”

Semiorthogonal decompositions and birational geometry of del Pezzo surfaces over arbitrary fields

“…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 is a smooth projective surface, Hassett and Tschinkel [, Lemma 8] prove that the index of can be recovered from as the greatest common divisor of the second Chern classes of objects.…”

“…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 ‐exceptional collection on a surface , 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.…”

Exceptional collections on Dolgachev surfaces associated with degenerations

“…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].…”