GIT stability of linear maps on projective space with marked points

Abstract

We construct moduli spaces of linear self-maps of P N with marked points, up to projective equivalence. That is, we let SL N +1 act simultaneously by conjugation on projective linear maps and diagonally on (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.

The algebraic dynamics of the pentagram map

“…Remark 3.7. The existence of the geometric quotient T n follows immediately from the main results of [47]. However, that proof does not produce the explicit coordinatization by corner invariants, and we use that coordinatization to study the pentagram map.…”

“…The construction of T n is not specific to the pentagram map and could have other applications. In fact, like Mumford's moduli space of closed polygons [29], the moduli space T n admits a GIT semistable compactification with an explicit combinatorial description [47].…”