Canonical tilting relative generators

Given a relatively projective birational morphism f:X \to Y of smooth algebraic spaces with dimension of fibers bounded by 1, we construct tilting relative (over Y) generators T_{X,f} and S_{X,f} in D^b(X). We develop a piece of general theory of strict admissible lattice filtrations in triangulated categories and show that D^b(X) has such a filtration L where the lattice is the set of all birational decompositions f: X\to Z \to Y with smooth Z. The t-structures related to T_{X,f} and S_{X,f} are proved to be glued via filtrations left and right dual to L. We realise all such Z as the fine moduli spaces of simple quotients of O_X in the heart of the t-structure for which S_{X,g} is a relative projective generator over Y. This implements the program of interpreting relevant smooth contractions of X in terms of a suitable system of t-structures on D^b(X).