American Institute of Mathematical Sciences

January  2016, 23: 25-40. doi: 10.3934/era.2016.23.004

Pentagrams, inscribed polygons, and Prym varieties

 1 Department of Mathematics, University of Toronto, Canada

Received  July 2016 Published  September 2016

The pentagram map is a discrete integrable system on the moduli space of planar polygons. The corresponding first integrals are so-called monodromy invariants $E_1, O_1, E_2, O_2,\dots$ 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 $E_k = O_k$ 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.
Citation: Anton Izosimov. Pentagrams, inscribed polygons, and Prym varieties. Electronic Research Announcements, 2016, 23: 25-40. doi: 10.3934/era.2016.23.004
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