In this talk I will introduce the notion of categorical representability, which relates an algebraic variety X to others, whose derived category appears inside a semi-orthogonal decomposition of D^b(X). I will outline the relations between this new definition and older ones, like algebraic representability and rational representability, and rationality properties of X. Then I will produce a couple of example where this notion is particularly relevant, focussing on the case of quadric fibrations and conic bundles. This is the outcome of joint projects with M. Bernardara.

Let X be an algebraic variety with Gorenstein singularities. A crepant resolution of X is often considered as some kind of minimal resolution of X. Unfortunately, varieties admitting such a resolution are rare. It is natural to consider "non commutative" resolution of singularities, and to find classes of varieties admitting non commutative crepant resolutions. In this talk we define the notion of wonderful resolution of singularities by analogy with the theory of wonderful compactification of linear algebraic groups. We sketch the proof of the following result: if a variety with rational singularities has a wonderful resolution of singularities, then it admits a non commutative crepant resolution. It is in particular the case of any determinantal variety, even symmetric or skew-symmetric. Finally we investigate the possibility of finding a non commutative crepant resolution in some cases where there is no wonderful resolution.

In this talk I will present a joint work with A.Auel and M.Bolognesi, in which we develop algebraic techniques such as relative homological projective duality for quadric pencils and Morita invariance of the even Clifford algebra under hyperbolic splitting. As an application, I will discuss a connection between rationality and semiorthogonal decompositions in some explicit examples in dimension 3 and 4.

I will report on the construction of an exceptional collection of maximal length 11 on the Barlow surface S. This can be used to show that in a small neighbourhood of S in the moduli space of determinantal Barlow surfaces, the generic surface has a semiorthogonal decomposition of its derived category into a length 11 exceptional sequence and a category with trivial Grothendieck group and Hochschild homology, called a phantom category. This is joint work with Christian Böhning, Hans-Christian Graf von Bothmer and Ludmil Katzarkov.

Given a smooth, positive-dimensional projective variety X, polarized by a very ample line bundle H, whose coordinate ring is Cohen-Macaulay, a particularly interesting class of sheaves on X is that of ACM bundles, namely E is such if H^i(E(t)) vanishes for all integers t and all 0 < i < dim(X). It turns out that there are few varieties admitting only finitely many indecomposable ACM bundles (up to a twist and isomorphism): projective spaces, smooth quadrics, rational normal curves, and two exceptional cases: the Veronese surface in P5 and the cubic scroll in P4. I will show that the Segre-Veronese embedding by O(d_1,...,d_s) of the product of s projective spaces of dimensions n_1,...,n_s supports families of arbitrarily high dimension of indecomposable ACM bundles (i.e. this variety is of "wild representation type") except for the above cases and for a the Segre embedding of a line and a smooth conic. This last variety is of "tame representation type", i.e. all families of indecomposable ACM bundles have dimension at most 1. This is the only known variety of this kind, besides the elliptic curve.

In this talk, I will use Kuznetsov’s description of the derived category of a smooth cubic hypersurface to give a new construction of some stable ACM bundles on cubic threefolds and cubic fourfolds containing a plane. This is a joint work with Emanuele Macrì and Paolo Stellari.