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(D, Y) as a 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 already 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. We are going to outline the relevance of the Auslander varieties in this setting.
Quiver Grassmannians, thus also Auslander varieties are projective varieties and it is known that every projective variety occurs as a quiver Grassmannian. There is a tendency to relate this fact to the wildness of quiver representations and the aim of the lecture is to clarify these thoughts: We show that for an algebra A which is (controlled) wild, any projective variety can be realized as an Auslander variety, but not necessarily as a quiver Grassmannian.