The paper deals with the following problem: Given a finitely generated associative algebra over an algebraically closed field, the algebraic variety of \(d\)-dimensional modules is endowed with a natural \(\text{Gl}_d\)-action. Various problems associated with this action are studied in the paper. Some strong results are obtained for algebras of finite representation type.
The author considers pointed varieties of the form \((\overline {O(m)},n)\) where \(\overline {O(m)}\) is the closure of the orbit of a module \(M\) and \(n\) is a minimal degeneration of \(M\) (see the paper for definitions). He calls the pointed varieties of the form \((\overline {O(m)}, n)\) minimal singularities.
The main theorem asserts that all minimal singularities occurring in representations of Dynkin quivers are very smoothly equivalent to \((D(p,q),0)\) where \(D(p,q)\) is the set of \(p \times q\) matrices with rank \(\leq 1\). The theorem at the end of section 1 relates degenerations of two distinct finite dimensional modules. It is a fundamental result used in the rest of the paper.
With this result the author derives the famous result of H. Kraft and C. Procesi on minimal singularities of conjugacy classes of nilpotent matrices [which appeared in Invent. Math. 53, 227-247 (1979; Zbl 0434.14026)] and states that in this setting any minimal singularity is equivalent to the subregular singularity inside the set of nilpotent matrices of some smaller size or to the singularity at 0 inside the set of all nilpotent matrices of rank at most one.
Section 4 deals with tilting modules, in particular corollary 1 shows a very close relation between the \(\text{Gl}_d\)-stable subsets of \(Y(\underline {d})\) and \(\text{Gl}_e\)-stable subsets of \(Y(\underline {e})\). Here a tilting module \(Y(\underline {d})\) consists of the category of torsion \(A\)-modules of vector dimension \(\underline {d}\). \(Y\) is the subcategory of \(B\)-mod corresponding to \(\tau\) (the torsion free part), and \(Y(\underline {e})\) is the full subcategory of \(Y\) whose objects have vector dimension \(\underline {e}\).
If \([M] = \underline {d}\) then \(\underline {e} = [\text{Hom} (T,M)] - [\text{Ext}^1 (T,M)]\). Theorem 3 and its corollary are very beautiful applications to tilting theory. Section 5 studies possible reduction of the underlying Gabriel quiver. Under reduction of the Gabriel quiver with some technical hypotheses the author gets associated pointed varieties which are very smoothly equivalent. Section 6 studies the minimal degeneration in the cases where the partial orders \(\leq\) and \(\leq_{\text{Ext}}\) are equivalent. This equivalence of the partial orders \(\leq_{\text{Ext}}\) and \(\leq\) is valid for preprojective modules; the author uses this fact and gets that any minimal degeneration of representations of a Dynkin quiver is of codimension one and then gets the main result which is Theorem 6. The proof of this result is very technical and complex.
The entire paper uses techniques of algebraic geometry applied to the representation theory of algebras. This paper is certainly a very nice one, although it requires from the reader a good knowledge of representation theory and algebraic geometry.