Canonical tilting relative generators

Given a relatively projective birational morphism f:X→Y of smooth algebraic spaces with dimension of fibers bounded by 1, we construct tilting relative (over Y) generators TX,f and SX,f in Db(X). We develop a piece of general theory of strict admissible lattice filtrations in triangulated categories and show that Db(X) has such a filtration L where the lattice is the set of all birational decompositions f:X→gZ→hY with smooth Z. The t-structures related to TX,f and SX,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 OX in the heart of the t-structure for which SX,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 Db(X).