Abstract. Let k be an algebraically closed field and A a finite-dimensional (associative) k-algebra. Given an A-module M, the set of all submodules of M with fixed dimension vector is called a quiver Grassmannian in M. If C, Y are A-modules, then we consider Hom(C, Y) as an E(C)-module, where E(C) is the opposite of the endomorphism ring of C, and the Auslander varieties for A are the quiver Grassmannians in this E(C)-module Hom(C,Y). In his seminal Philadelphia Notes (published in 1978), M. Auslander exhibited his theory of morphisms determined by modules. It is an important frame for describing the poset structure of the category of A-modules and this setting allows to interpret the Auslander varieties as describing factorizations of morphisms of A-modules.
If the algebra A is (controlled) wild, then one knows that any projective variety can be realized as an Auslander variety. The aim of the lecture is to analyse sets of factorizations which can be used to realize arbitrary projective varieties.