Let X → Y be a fibration whose fibers are complete intersections of r quadrics. We develop new categorical and algebraic tools-a theory of relative homological projective duality and the Morita invariance of the even Clifford algebra under quadric reduction by hyperbolic splitting-to study semiorthogonal decompositions of the bounded derived category D b (X). Together with results in the theory of quadratic forms, we apply these tools in the case where r = 2 and X → Y has relative dimension 1, 2, or 3, in which case the fibers are curves of genus one, Del Pezzo surfaces of degree 4, or Fano threefolds, respectively. In the latter two cases, if Y = P 1 over an algebraically closed field of characteristic zero, we relate rationality questions to categorical representability of X.

We isolate a class of smooth rational cubic fourfolds X containing a plane whose associated quadric surface bundle does not have a rational section. This is equivalent to the nontriviality of the Brauer class β of the even Clifford algebra over the K3 surface S of degree 2 arising from X. Specifically, we show that in the moduli space of cubic fourfolds, the intersection of divisors C 8 ∩ C 14 has five irreducible components. In the component corresponding to the existence of a tangent conic, we prove that the general member is both pfaffian and has β nontrivial. Such cubic fourfolds provide twisted derived equivalences between K3 surfaces of degrees 2 and 14, hence further corroboration of Kuznetsov's derived categorical conjecture on the rationality of cubic fourfolds.

In this paper we investigate the divisor C14 inside the moduli space of smooth cubic hypersurfaces in P 5 , whose general element is a smooth cubic containing a smooth quartic rational normal scroll. By showing that all degenerations of quartic scrolls in P 5 contained in a smooth cubic hypersurface are surfaces with one apparent double point, we prove that every cubic hypersurface contained in C14 is rational. Combining our proof with the Hodge theoretic definition of C14, we deduce that on a smooth cubic fourfold every class T ∈ H 2,2 (X, Z) with T 2 = 10 and T · h 2 = 4 is represented by a (possibly reducible) surface of degree four which has one apparent double point. As an application of our results and of the construction of some explicit examples, we also prove that the Pfaffian locus is not open in C14.

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.

International audienceIn this paper we prove Homological Projective Duality for categorical resolutions of several classes of linear determinantal varieties. By this we mean varieties that are cut out by the minors of a given rank of a m x n matrix of linear forms on a given projective space. As applications, we obtain pairs of derived-equivalent Calabi-Yau manifolds, and address a question by A. Bondal asking whether the derived category of any smooth projective variety can be fully faithfully embedded in the derived category of a smooth Fano variety. Moreover we, discuss the relation between rationality and categorical representability in codimension two for determinantal varieties. (C) 2016 Elsevier Inc. All rights reserved

Abstract. We define, basing upon semiorthogonal decompositions of D b (X), categorical representability of a projective variety X and describe its relation with classical representabilities of the Chow ring. For complex threefolds satisfying both classical and categorical representability assumptions, we reconstruct the intermediate Jacobian from the semiorthogonal decomposition. We discuss finally how categorical representability can give useful information on the birational properties of X by providing examples and stating open questions.

In this paper we study the stack Tg of smooth triple covers of a conic; when g ≥ 5 this stack is embedded Mg as the locus of trigonal curves. We show that Tg is a quotient [Ug/Γg ], where Γg is a certain algebraic group and Ug is an open subscheme of a Γg-equivariant vector bundle over an open subscheme of a representation of Γg . Using this, we compute the integral Picard group of Tg when g > 1. The main tools are a result of Miranda that describes a flat finite triple cover of a scheme S as given by a locally free sheaf E of rank two on S, with a section of Sym 3 E ⊗ det E ∨ , and a new description of the stack of globally generated locally free sheaves of fixed rank and degree on a projective line as a quotient stack.

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