GIT stability of linear maps on projective space with marked points

TL;DR

This paper constructs moduli spaces of linear maps on projective space with marked points using GIT, providing a combinatorial criterion for stability based on counting points on specific flags.

Contribution

It introduces a new GIT stability criterion for linear maps with marked points, linking stability to combinatorial counts on flags with Hessenberg functions.

Findings

GIT stability characterized by counting marked points on flags

Explicit description of weight polytopes via convex polyhedra

Moduli spaces relevant to algebraic dynamics and integrable systems

Abstract

We construct moduli spaces of linear self-maps of projective space with marked points, up to projective equivalence. That is, we let the special linear group act simultaneously by conjugation on projective linear maps and diagonally on $(\mathbb{P}^N)^n$, and we take the geometric invariant theory (GIT) quotient. These moduli spaces arise in algebraic dynamics and integrable systems. Our main result is a dynamical characterization of the GIT semistable and stable loci in the space of linear maps $T$ with marked points. We show that GIT stability can be checked by counting the marked points on flags with certain Hessenberg functions relative to $T$. The proof is combinatorial: to describe the weight polytopes for this action, we compute the vertices and facets of certain convex polyhedra generated by roots of the $A_N$ lattice.