GIT stability of linear maps on projective space with marked points

Max Weinreich

Published 2021-11-11Version 1

We construct moduli spaces of linear self-maps of $N$-dimensional projective space with $n$ 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 in two ways: first, as ambient varieties of degree 1 portrait spaces; second, as the domains of discrete integrable systems such as the pentagram map. Our main result is a dynamical characterization of the GIT semistable and stable loci in the space of linear maps with marked points. 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.