Derived categories and rationality of conic bundles

Abstract

Abstract. We show that a standard conic bundle over a minimal rational surface is rational and its Jacobian splits as the direct sum of Jacobians of curves if and only if its derived category admits a semiorthogonal decomposition by exceptional objects and the derived categories of those curves. Moreover, such a decomposition gives the splitting of the intermediate Jacobian also when the surface is not minimal.

Categorical representability and intermediate Jacobians of Fano threefolds

“…A positive answer to the second question is provided for standard conic bundles over minimal surfaces [BB10], but it seems to be quite a strong fact to hold in general: recall that having a splitting J(X) ≃ ⊕J(Γ i ) is only a necessary condition for rationality, and Corollary 3.7 shows that if X satisfies ♮, categorical representability in codimension 2 would give the splitting of the Jacobian.…”

“…The categorical representability of (S, B 0 ) should then be a very important tool in studying birational properties of Q. This is indeed the case for conic bundles over rational surfaces [BB10]. Finally, let A be an 4.4.…”

“…Higher dimensional varieties. Unfortunately, it looks like the techniques used for threefolds in [BB10] hardly extend to dimension bigger than 3. The examples and supporting evidences provided so far lead anyway to suppose that categorical representability can give useful informations on the birational properties of any projective variety.…”

Homological projective duality for determinantal varieties

“…Several examples seem to suggest that Question 1.2 may have a positive answer. Let us mention conic bundles over minimal surfaces [6], fibrations in intersections of quadrics [3], or some classes of cubic fourfolds [23]. Moreover, Question 1.2 is equivalent to one implication of Kuznetsov Conjecture on the rationality of a cubic fourfold [23], which was proved to coincide with Hodge theoretical expectations for a general cubic fourfold by Addington and Thomas [1].…”

Cubic fourfolds containing a plane and a quintic del Pezzo surface

“…First, while the locus of pfaffian cubic fourfolds is dense in C 14 , it is not true that the locus of pfaffians containing a plane is dense in (all components of) C 8 ∩ C 14 . As for the general study of quadric bundles over surfaces, there do exist conic bundles over surfaces, without rational sections, whose total space is smooth projective rational: these are classified over rational minimal surfaces [Sho84] (see also [BB13]); over P 2 their discriminant curves have degree at most five [Bea77]. However, there is no analogous classification of quadric surface bundles over surfaces with smooth projective rational total space and without rational sections.…”