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| Format: | Preprint |
| Published: |
2021
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2111.06351 |
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| _version_ | 1866913425545756672 |
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| author | Weinreich, Max |
| author_facet | Weinreich, Max |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_06351 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | GIT stability of linear maps on projective space with marked points Weinreich, Max Algebraic Geometry Dynamical Systems 37P45, 14L24, 52B05 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. |
| title | GIT stability of linear maps on projective space with marked points |
| topic | Algebraic Geometry Dynamical Systems 37P45, 14L24, 52B05 |
| url | https://arxiv.org/abs/2111.06351 |