Quivers, Geometric Invariant Theory, and Moduli of Linear Dynamical Systems

TL;DR

This paper employs geometric invariant theory and quiver representations to analyze and describe compactifications of moduli spaces of linear dynamical systems, revealing their geometric structures and differences.

Contribution

It introduces a unified approach to compactify moduli spaces using GIT and quivers, and characterizes two known compactifications as Quot schemes and Grassmann bundles.

Findings

Both compactifications are smooth projective manifolds.

Lomadze's compactification is described as a Quot scheme.

Helmke's compactification is an algebraic Grassmann bundle over a Quot scheme.

Abstract

We use geometric invariant theory and the language of quivers to study compactifications of moduli spaces of linear dynamical systems. A general approach to this problem is presented and applied to two well known cases: We show how both Lomadze's and Helmke's compactification arises naturally as a geometric invariant theory quotient. Both moduli spaces are proven to be smooth projective manifolds. Furthermore, a description of Lomadze's compactification as a Quot scheme is given, whereas Helmke's compactification is shown to be an algebraic Grassmann bundle over a Quot scheme. This gives an algebro-geometric description of both compactifications. As an application, we determine the cohomology ring of Helmke's compactification and prove that the two compactifications are not isomorphic when the number of outputs is positive.