Pentagrams, inscribed polygons, and Prym varieties

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

Pentagrams, inscribed polygons, and Prym varieties

Anton Izosimov ^1,

1.	Department of Mathematics, University of Toronto, Canada

Received July 2016 Published September 2016

Abstract
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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.

Keywords: Pentagram map, conic sections, Prym varieties..

Mathematics Subject Classification: 35K55, 35R4.

Citation: Anton Izosimov. Pentagrams, inscribed polygons, and Prym varieties. Electronic Research Announcements, 2016, 23: 25-40. doi: 10.3934/era.2016.23.004

References:

[1]	R. Felipe and G. Marí Beffa, The pentagram map on Grassmannians, to appear in Ann. Inst. Fourier, arXiv:1507.04765, (2015).
[2]	V. V. Fock and A. Marshakov, Loop groups, clusters, dimers and integrable systems, arXiv:1401.1606, (2014).
[3]	D. Fuchs and S. Tabachnikov, Self-dual polygons and self-dual curves, Funct. Anal. Other Math., 2 (2009), 203-220. doi: 10.1007/s11853-008-0020-5.
[4]	M. Gekhtman, M. Shapiro, S. Tabachnikov and A. Vainshtein, Integrable cluster dynamics of directed networks and pentagram maps, with an appendix by A. Izosimov, Adv. Math., 300 (2016), 390-450. doi: 10.1016/j.aim.2016.03.023.
[5]	M. Glick, The pentagram map and Y-patterns, Adv. Math., 227 (2011), 1019-1045. doi: 10.1016/j.aim.2011.02.018.
[6]	M. Glick, The Devron property, J. Geom. Phys., 87 (2015), 161-189. doi: 10.1016/j.geomphys.2014.07.029.
[7]	M. Glick and P. Pylyavskyy, Y-meshes and generalized pentagram maps, Proc. London Math. Soc. (3), 112 (2016), 753-797. doi: 10.1112/plms/pdw007.
[8]	R. Kedem and P. Vichitkunakorn, T-systems and the pentagram map, J. Geom. Phys., 87 (2015), 233-247. doi: 10.1016/j.geomphys.2014.07.003.
[9]	B. Khesin and F. Soloviev, Integrability of higher pentagram maps, Math. Ann., 357 (2013), 1005-1047. doi: 10.1007/s00208-013-0922-5.
[10]	B. Khesin and F. Soloviev, The geometry of dented pentagram maps, J. Eur. Math. Soc., 18 (2016), 147-179. doi: 10.4171/JEMS/586.
[11]	G. Marí Beffa, On integrable generalizations of the pentagram map, Int. Math. Res. Not., (2015), 3669-3693. doi: 10.1093/imrn/rnu044.
[12]	V. Ovsienko, R. Schwartz and S. Tabachnikov, The pentagram map: A discrete integrable system, Comm. Math. Phys., 299 (2010), 409-446. doi: 10.1007/s00220-010-1075-y.
[13]	V. Ovsienko, R. Schwartz and S. Tabachnikov, Liouville-Arnold integrability of the pentagram map on closed polygons, Duke Math. J., 162 (2013), 2149-2196. doi: 10.1215/00127094-2348219.
[14]	R. Schwartz, The pentagram map, Exp. Math., 1 (1992), 71-81.
[15]	R. Schwartz, The pentagram map is recurrent, Exp. Math., 10 (2001), 519-528. doi: 10.1080/10586458.2001.10504671.
[16]	R. Schwartz, The Poncelet grid, Adv. Geom., 7 (2007), 157-175. doi: 10.1515/ADVGEOM.2007.010.
[17]	R. Schwartz, Discrete monodromy, pentagrams, and the method of condensation, J. Fixed Point Theory Appl., 3 (2008), 379-409. doi: 10.1007/s11784-008-0079-0.
[18]	R. Schwartz, The pentagram integrals for Poncelet families, J. Geom. Phys., 87 (2015), 432-449. doi: 10.1016/j.geomphys.2014.07.024.
[19]	R. Schwartz and S. Tabachnikov, Elementary surprises in projective geometry, Math. Intelligencer, 32 (2010), 31-34. doi: 10.1007/s00283-010-9137-8.
[20]	R. Schwartz and S. Tabachnikov, The pentagram integrals on inscribed polygons, Electron. J. Combin., 18 (2011), Paper 171, 19pp.
[21]	F. Soloviev, Integrability of the pentagram map, Duke Math. J., 162 (2013), 2815-2853. doi: 10.1215/00127094-2382228.

show all references

References:

[1]	R. Felipe and G. Marí Beffa, The pentagram map on Grassmannians, to appear in Ann. Inst. Fourier, arXiv:1507.04765, (2015).
[2]	V. V. Fock and A. Marshakov, Loop groups, clusters, dimers and integrable systems, arXiv:1401.1606, (2014).
[3]	D. Fuchs and S. Tabachnikov, Self-dual polygons and self-dual curves, Funct. Anal. Other Math., 2 (2009), 203-220. doi: 10.1007/s11853-008-0020-5.
[4]	M. Gekhtman, M. Shapiro, S. Tabachnikov and A. Vainshtein, Integrable cluster dynamics of directed networks and pentagram maps, with an appendix by A. Izosimov, Adv. Math., 300 (2016), 390-450. doi: 10.1016/j.aim.2016.03.023.
[5]	M. Glick, The pentagram map and Y-patterns, Adv. Math., 227 (2011), 1019-1045. doi: 10.1016/j.aim.2011.02.018.
[6]	M. Glick, The Devron property, J. Geom. Phys., 87 (2015), 161-189. doi: 10.1016/j.geomphys.2014.07.029.
[7]	M. Glick and P. Pylyavskyy, Y-meshes and generalized pentagram maps, Proc. London Math. Soc. (3), 112 (2016), 753-797. doi: 10.1112/plms/pdw007.
[8]	R. Kedem and P. Vichitkunakorn, T-systems and the pentagram map, J. Geom. Phys., 87 (2015), 233-247. doi: 10.1016/j.geomphys.2014.07.003.
[9]	B. Khesin and F. Soloviev, Integrability of higher pentagram maps, Math. Ann., 357 (2013), 1005-1047. doi: 10.1007/s00208-013-0922-5.
[10]	B. Khesin and F. Soloviev, The geometry of dented pentagram maps, J. Eur. Math. Soc., 18 (2016), 147-179. doi: 10.4171/JEMS/586.
[11]	G. Marí Beffa, On integrable generalizations of the pentagram map, Int. Math. Res. Not., (2015), 3669-3693. doi: 10.1093/imrn/rnu044.
[12]	V. Ovsienko, R. Schwartz and S. Tabachnikov, The pentagram map: A discrete integrable system, Comm. Math. Phys., 299 (2010), 409-446. doi: 10.1007/s00220-010-1075-y.
[13]	V. Ovsienko, R. Schwartz and S. Tabachnikov, Liouville-Arnold integrability of the pentagram map on closed polygons, Duke Math. J., 162 (2013), 2149-2196. doi: 10.1215/00127094-2348219.
[14]	R. Schwartz, The pentagram map, Exp. Math., 1 (1992), 71-81.
[15]	R. Schwartz, The pentagram map is recurrent, Exp. Math., 10 (2001), 519-528. doi: 10.1080/10586458.2001.10504671.
[16]	R. Schwartz, The Poncelet grid, Adv. Geom., 7 (2007), 157-175. doi: 10.1515/ADVGEOM.2007.010.
[17]	R. Schwartz, Discrete monodromy, pentagrams, and the method of condensation, J. Fixed Point Theory Appl., 3 (2008), 379-409. doi: 10.1007/s11784-008-0079-0.
[18]	R. Schwartz, The pentagram integrals for Poncelet families, J. Geom. Phys., 87 (2015), 432-449. doi: 10.1016/j.geomphys.2014.07.024.
[19]	R. Schwartz and S. Tabachnikov, Elementary surprises in projective geometry, Math. Intelligencer, 32 (2010), 31-34. doi: 10.1007/s00283-010-9137-8.
[20]	R. Schwartz and S. Tabachnikov, The pentagram integrals on inscribed polygons, Electron. J. Combin., 18 (2011), Paper 171, 19pp.
[21]	F. Soloviev, Integrability of the pentagram map, Duke Math. J., 162 (2013), 2815-2853. doi: 10.1215/00127094-2382228.

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