Pavings of polyhedra, glueing of Schubert cells and compactification of configuration spaces

A lecture in projective geometry. When studying the compactifications of Drinfeld's moduli spaces of shtukas with level structure or (according to Faltings) local models of Shimura varieties, one is led to the problem of compactifying the quotients $PGL(r)\times\dots\times PGL(r)/PGL(r)$ in an equivariant way. A general method for compactifying these quotients is presented. It also applies to configuration spaces of matroids. All the compactified schemes we obtain are endowed with a structure morphism (which is smooth when there are at most 3 factors or when the rank is 2 but not in general) over a “toric stack” whose points are the pavings of some integral polyhedron. There is an induced stratification and the strata can be described in terms of glueing of thin Schubert cells. And all the compactified schemes have at least two modular interpretations:
- classifying equivariant vector bundles on some toric varieties,
- classifying some kind of projective rational varieties with logarithmic singularities (which generalise the “minimal models of projective spaces” introduced by Faltings).