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
References:
[1]

R. Felipe and G. Marí Beffa, The pentagram map on Grassmannians,, to appear in Ann. Inst. Fourier, (2015). Google Scholar

[2]

V. V. Fock and A. Marshakov, Loop groups, clusters, dimers and integrable systems,, , (2014). Google Scholar

[3]

D. Fuchs and S. Tabachnikov, Self-dual polygons and self-dual curves,, Funct. Anal. Other Math., 2 (2009), 203. doi: 10.1007/s11853-008-0020-5. Google Scholar

[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, 300 (2016), 390. doi: 10.1016/j.aim.2016.03.023. Google Scholar

[5]

M. Glick, The pentagram map and Y-patterns,, Adv. Math., 227 (2011), 1019. doi: 10.1016/j.aim.2011.02.018. Google Scholar

[6]

M. Glick, The Devron property,, J. Geom. Phys., 87 (2015), 161. doi: 10.1016/j.geomphys.2014.07.029. Google Scholar

[7]

M. Glick and P. Pylyavskyy, Y-meshes and generalized pentagram maps,, Proc. London Math. Soc. (3), 112 (2016), 753. doi: 10.1112/plms/pdw007. Google Scholar

[8]

R. Kedem and P. Vichitkunakorn, T-systems and the pentagram map,, J. Geom. Phys., 87 (2015), 233. doi: 10.1016/j.geomphys.2014.07.003. Google Scholar

[9]

B. Khesin and F. Soloviev, Integrability of higher pentagram maps,, Math. Ann., 357 (2013), 1005. doi: 10.1007/s00208-013-0922-5. Google Scholar

[10]

B. Khesin and F. Soloviev, The geometry of dented pentagram maps,, J. Eur. Math. Soc., 18 (2016), 147. doi: 10.4171/JEMS/586. Google Scholar

[11]

G. Marí Beffa, On integrable generalizations of the pentagram map,, Int. Math. Res. Not., (2015), 3669. doi: 10.1093/imrn/rnu044. Google Scholar

[12]

V. Ovsienko, R. Schwartz and S. Tabachnikov, The pentagram map: A discrete integrable system,, Comm. Math. Phys., 299 (2010), 409. doi: 10.1007/s00220-010-1075-y. Google Scholar

[13]

V. Ovsienko, R. Schwartz and S. Tabachnikov, Liouville-Arnold integrability of the pentagram map on closed polygons,, Duke Math. J., 162 (2013), 2149. doi: 10.1215/00127094-2348219. Google Scholar

[14]

R. Schwartz, The pentagram map,, Exp. Math., 1 (1992), 71. Google Scholar

[15]

R. Schwartz, The pentagram map is recurrent,, Exp. Math., 10 (2001), 519. doi: 10.1080/10586458.2001.10504671. Google Scholar

[16]

R. Schwartz, The Poncelet grid,, Adv. Geom., 7 (2007), 157. doi: 10.1515/ADVGEOM.2007.010. Google Scholar

[17]

R. Schwartz, Discrete monodromy, pentagrams, and the method of condensation,, J. Fixed Point Theory Appl., 3 (2008), 379. doi: 10.1007/s11784-008-0079-0. Google Scholar

[18]

R. Schwartz, The pentagram integrals for Poncelet families,, J. Geom. Phys., 87 (2015), 432. doi: 10.1016/j.geomphys.2014.07.024. Google Scholar

[19]

R. Schwartz and S. Tabachnikov, Elementary surprises in projective geometry,, Math. Intelligencer, 32 (2010), 31. doi: 10.1007/s00283-010-9137-8. Google Scholar

[20]

R. Schwartz and S. Tabachnikov, The pentagram integrals on inscribed polygons,, Electron. J. Combin., 18 (2011). Google Scholar

[21]

F. Soloviev, Integrability of the pentagram map,, Duke Math. J., 162 (2013), 2815. doi: 10.1215/00127094-2382228. Google Scholar

show all references

References:
[1]

R. Felipe and G. Marí Beffa, The pentagram map on Grassmannians,, to appear in Ann. Inst. Fourier, (2015). Google Scholar

[2]

V. V. Fock and A. Marshakov, Loop groups, clusters, dimers and integrable systems,, , (2014). Google Scholar

[3]

D. Fuchs and S. Tabachnikov, Self-dual polygons and self-dual curves,, Funct. Anal. Other Math., 2 (2009), 203. doi: 10.1007/s11853-008-0020-5. Google Scholar

[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, 300 (2016), 390. doi: 10.1016/j.aim.2016.03.023. Google Scholar

[5]

M. Glick, The pentagram map and Y-patterns,, Adv. Math., 227 (2011), 1019. doi: 10.1016/j.aim.2011.02.018. Google Scholar

[6]

M. Glick, The Devron property,, J. Geom. Phys., 87 (2015), 161. doi: 10.1016/j.geomphys.2014.07.029. Google Scholar

[7]

M. Glick and P. Pylyavskyy, Y-meshes and generalized pentagram maps,, Proc. London Math. Soc. (3), 112 (2016), 753. doi: 10.1112/plms/pdw007. Google Scholar

[8]

R. Kedem and P. Vichitkunakorn, T-systems and the pentagram map,, J. Geom. Phys., 87 (2015), 233. doi: 10.1016/j.geomphys.2014.07.003. Google Scholar

[9]

B. Khesin and F. Soloviev, Integrability of higher pentagram maps,, Math. Ann., 357 (2013), 1005. doi: 10.1007/s00208-013-0922-5. Google Scholar

[10]

B. Khesin and F. Soloviev, The geometry of dented pentagram maps,, J. Eur. Math. Soc., 18 (2016), 147. doi: 10.4171/JEMS/586. Google Scholar

[11]

G. Marí Beffa, On integrable generalizations of the pentagram map,, Int. Math. Res. Not., (2015), 3669. doi: 10.1093/imrn/rnu044. Google Scholar

[12]

V. Ovsienko, R. Schwartz and S. Tabachnikov, The pentagram map: A discrete integrable system,, Comm. Math. Phys., 299 (2010), 409. doi: 10.1007/s00220-010-1075-y. Google Scholar

[13]

V. Ovsienko, R. Schwartz and S. Tabachnikov, Liouville-Arnold integrability of the pentagram map on closed polygons,, Duke Math. J., 162 (2013), 2149. doi: 10.1215/00127094-2348219. Google Scholar

[14]

R. Schwartz, The pentagram map,, Exp. Math., 1 (1992), 71. Google Scholar

[15]

R. Schwartz, The pentagram map is recurrent,, Exp. Math., 10 (2001), 519. doi: 10.1080/10586458.2001.10504671. Google Scholar

[16]

R. Schwartz, The Poncelet grid,, Adv. Geom., 7 (2007), 157. doi: 10.1515/ADVGEOM.2007.010. Google Scholar

[17]

R. Schwartz, Discrete monodromy, pentagrams, and the method of condensation,, J. Fixed Point Theory Appl., 3 (2008), 379. doi: 10.1007/s11784-008-0079-0. Google Scholar

[18]

R. Schwartz, The pentagram integrals for Poncelet families,, J. Geom. Phys., 87 (2015), 432. doi: 10.1016/j.geomphys.2014.07.024. Google Scholar

[19]

R. Schwartz and S. Tabachnikov, Elementary surprises in projective geometry,, Math. Intelligencer, 32 (2010), 31. doi: 10.1007/s00283-010-9137-8. Google Scholar

[20]

R. Schwartz and S. Tabachnikov, The pentagram integrals on inscribed polygons,, Electron. J. Combin., 18 (2011). Google Scholar

[21]

F. Soloviev, Integrability of the pentagram map,, Duke Math. J., 162 (2013), 2815. doi: 10.1215/00127094-2382228. Google Scholar

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