The Algebraic Dynamics of the Pentagram Map

Weinreich, Max

doi:10.48550/arxiv.2104.06211

Cited by 3 publications

(4 citation statements)

References 29 publications

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“…In previous work, we showed that, in the planar case (N = 2), if n ≥ 4, the moduli space M 2 1,n is birational to P 2n , by working in explicit coordinates [22,Theorem 3.2]. This generalizes:…”

Section: Introductionmentioning

confidence: 76%

“…Note that the SL 3 -equivalence class of the monodromy T is an invariant of the pentagram map, providing two algebraically independent conserved quantities. In fact, the pentagram map is a discrete algebraic completely integrable system: it has an iterate which is birational to a translation on a family of abelian varieties [21,22]. But this is only a description of the generic behavior of the map.…”

Section: Introductionmentioning

confidence: 99%

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GIT stability of linear maps on projective space with marked points

Weinreich¹

2021

Preprint

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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.

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Section: Introductionmentioning

confidence: 76%

Section: Introductionmentioning

confidence: 99%

GIT stability of linear maps on projective space with marked points

Weinreich¹

2021

Preprint

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“…The pentagram map, T 2 , is one of the best known discrete completely integrable system. See for instance, [1], [2], [3], [8], [9], [10], [11], [12], [14], [15], [21], [22]. The Pentagram Rigidity Conjecture would be a statement about the global topology of an abelian foliation associated to a completely integrable system if the map T 3 were known to be completely integrable.…”

Section: Contextmentioning

confidence: 99%

Pentagram Rigidity for Centrally Symmetric Octagons

Schwartz¹

2021

Preprint

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In this paper I will establish a special case of a conjecture that intertwines the deep diagonal pentagram maps and Poncelet polygons. The special case is that of the 3-diagonal map acting on affine equivalence classes of centrally symmetric octagons. This is the simplest case that goes beyond an analysis of elliptic curves. The proof involves establishing that the map is Arnold-Liouville integrable in this case, and then exploring the Lagrangian surface foliation in detail.

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“…This implies in particular that the torus foliation discussed above is naturally an abelian fibration, with the individual tori having natural desciptions as Jacobian varieties for certain Riemann surfaces. Very recently, M. Weinreich [19] proved that T 2 is algebro-geometrically integrable in any field of characteristic not equal to 2. In [3], M. Glick related the pentagram map to a cluster algebra.…”

Section: Contextmentioning

confidence: 99%