Moduli spaces for endomorphisms of finite dimensional vector spaces (Q1275298)

The author considers pairs \((V,T)\) where \(V\) is a \(n\)-dimensional vector space over a field \(K\), not necessarily algebraically closed, and \(T\) is an endomorphism of \(V\). The moduli problem for pairs \((V,T)\) according to the obvious equivalence relation is called \((End)_n\). \(T\in End(V)\) is called cyclic if \(V\) is a principal \(K[T]\)-module. It is well known that \((End)_n\) has no coarse moduli space, while if we consider only the cyclic endomorphisms Mumford has shown that we get a fine moduli space. The author proves that \((End)_n\) has a natural stratification according to the dimension of the \(T\)-cyclic decomposition of \(V\), such that each stratum has a fine moduli space. The bigger stratum consists of cyclic endomorphisms.