Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

GIT stability of linear maps on projective space with marked points

Max Weinreich

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.

GIT stability of linear maps on projective space with marked points

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 , 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 with marked points. We show that GIT stability can be checked by counting the marked points on flags with certain Hessenberg functions relative to . 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 lattice.

Paper Structure

This paper contains 14 sections, 26 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Sketch of the proof of Theorem \ref{['thm_main']}
  3. Motivation from dynamical moduli spaces
  4. Motivation from combinatorial algebraic geometry
  5. Motivation from integrable systems
  6. Outline
  7. Preliminaries
  8. Corner polyhedra and stability
  9. First application: configurations of points
  10. Corner polyhedra of type A N
  11. Facets of corner polyhedra of type A N
  12. Examples
  13. Linear maps with marked points
  14. Application: one marked point

Key Result

Theorem 1.3

Let $k$ be an algebraically closed field, and let $N, n, q \in \mathbb N$. We consider the variety equipped with the action eq_acn of $\mathop{\mathrm{SL}}\nolimits_{N+1}$, the sheaf $\mathcal{L} = \mathcal{O}(q, 1, \hdots, 1)$, and the unique $\mathop{\mathrm{SL}}\nolimits_{N+1}$-linearization.

Theorems & Definitions (79)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Example 1.4
  • Example 1.5
  • Example 1.6
  • Example 1.7
  • Corollary 1.8
  • Theorem 1.9
  • Definition 1.11
  • ...and 69 more