A derived category approach to some moduli spaces on cubic threefolds and fourfolds.

We exploit the homological properties and the geometric meaning of
Kuznetsov's semiorthogonal decomposition of the derived categories of
cubic fourfolds (and threefolds) to study the (birational) geometry of
some interesting moduli spaces on such varieties. We will start
working out the very instructive example of the moduli space of stable
aCM bundles of a given rank on a cubic threefold. Then we will discuss
our main result concerning the case of generalized twisted cubics on
cubic fourfolds not containing a plane. We will show that we can
recover the picture by Lehn-Lehn-Sorger-van Straten in terms of moduli
spaces of (weakly) stable sheaves/complexes. This is joint work with
M. Lahoz, M. Lehn, and E. Macri'.