Irreducible components of varieties of representations: The local case

For any positive integer d, we determine the irreducible components of the varieties that parametrize the d  -dimensional representations of a local truncated path algebra Λ. Here Λ is a quotient KQ/〈the paths of lengthL+1〉 of a path algebra KQ, where K is an algebraically closed field, L is a positive integer, and Q is the quiver with a single vertex and a finite number r   of loops. The components are determined in both the classical and the Grassmannian settings, Repd(Λ)Repd(Λ) and GRASSd(Λ)GRASSd(Λ). Our method is to corner the components by way of a twin pair of upper semicontinuous maps from Repd(Λ)Repd(Λ) to a poset consisting of sequences of semisimple modules.An excerpt of the main result is as follows. Given a sequence S=(S0,…,SL)S=(S0,…,SL) of semisimple modules with dim⁡⨁0≤l≤LSl=ddim⁡⨁0≤l≤LSl=d, let RepS be the subvariety of Repd(Λ)Repd(Λ) consisting of the points that parametrize the modules with radical layering SS. (The radical layering of a Λ-module M   is the sequence (JlM/Jl+1M)0≤l≤L(JlM/Jl+1M)0≤l≤L, where J is the Jacobson radical of Λ.) Suppose the quiver Q   has r≥2r≥2 loops. If d≤L+1d≤L+1, the variety Repd(Λ)Repd(Λ) is irreducible and, generically, its modules are uniserial. If, on the other hand, d>L+1d>L+1, then the irreducible components of Repd(Λ)Repd(Λ) are the closures of the subvarieties RepS for those sequences SS which satisfy the inequalities dim⁡Sl≤r⋅dim⁡Sl+1dim⁡Sl≤r⋅dim⁡Sl+1 and dim⁡Sl+1≤r⋅dim⁡Sldim⁡Sl+1≤r⋅dim⁡Sl for 0≤l