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The pentagram map is a discrete integrable system on the moduli space ofplanar polygons. The corresponding first integrals are so-called monodromyinvariants $E_1, O_1, E_2, O_2,\dots$ By analyzing the combinatorics of theseinvariants, R.Schwartz and S.Tabachnikov have recently proved that for polygonsinscribed in a conic section one has $E_k = O_k$ for all $k$. In this paper wegive a simple conceptual proof of the Schwartz-Tabachnikov theorem. Our mainobservation is that for inscribed polygons the corresponding monodromysatisfies a certain self-duality relation. From this we also deduce that thespace of inscribed polygons with fixed values of the monodromy invariants is anopen dense subset in the Prym variety (i.e., a half-dimensional torus in theJacobian) of the spectral curve. As a byproduct, we also prove anotherconjecture of Schwartz and Tabachnikov on positivity of monodromy invariantsfor convex polygons.

Pentagrams, inscribed polygons, and Prym varieties