Pentagrams, inscribed polygons, and prym varieties

Research output: Contribution to journalArticle

Abstract

The pentagram map is a discrete integrable system on the module space of planar polygons. The correspondingrst integrals are so-called monodromy invariants E1;O1;E2;O2;… By analyzing the combinatorics of these invariants, R. Schwartz and S. Tabachnikov have recently proved that for polygons inscribed in a conic section one has Ek = Ok for all k. In this paper we give a simple conceptual proof of the Schwartz-Tabachnikov theorem. Our main observation is that for inscribed polygons the corresponding monodromy satisfies a certain self-duality relation. From this we also deduce that the space of inscribed polygons with Fixed values of the monodromy invariants is an open dense subset in the Prym variety (i.e., a half-dimensional torus in the Jacobian) of the spectral curve. As a byproduct, we also prove another conjecture of Schwartz and Tabachnikov on positivity of monodromy invariants for convex polygons.

Original languageEnglish (US)
Pages (from-to)25-40
Number of pages16
JournalElectronic Research Announcements in Mathematical Sciences
Volume23
DOIs
StatePublished - Jan 30 2016
Externally publishedYes

Keywords

  • Conic sections
  • Pentagram map
  • Prym varieties

ASJC Scopus subject areas

  • Mathematics(all)

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