American Institute of Mathematical Sciences

Advanced
  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

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, (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

[1]

Valentin Ovsienko, Richard Schwartz, Serge Tabachnikov. Quasiperiodic motion for the pentagram map. Electronic Research Announcements, 2009, 16: 1-8. doi: 10.3934/era.2009.16.1
[2]

Boris Khesin, Fedor Soloviev. The pentagram map in higher dimensions and KdV flows. Electronic Research Announcements, 2012, 19: 86-96. doi: 10.3934/era.2012.19.86
[3]

Eric Férard. On the irreducibility of the hyperplane sections of Fermat varieties in $\mathbb{P}^3$ in characteristic $2$. Advances in Mathematics of Communications, 2014, 8 (4) : 497-509. doi: 10.3934/amc.2014.8.497
[4]

V. Kumar Murty, Ying Zong. Splitting of abelian varieties. Advances in Mathematics of Communications, 2014, 8 (4) : 511-519. doi: 10.3934/amc.2014.8.511
[5]

Daniel Grieser. A natural differential operator on conic spaces. Conference Publications, 2011, 2011 (Special) : 568-577. doi: 10.3934/proc.2011.2011.568
[6]

G. Mashevitzky, B. Plotkin and E. Plotkin. Automorphisms of categories of free algebras of varieties. Electronic Research Announcements, 2002, 8: 1-10.

[7]

A. Giambruno and M. Zaicev. Minimal varieties of algebras of exponential growth. Electronic Research Announcements, 2000, 6: 40-44.

[8]

Laurenţiu Maxim, Jörg Schürmann. Characteristic classes of singular toric varieties. Electronic Research Announcements, 2013, 20: 109-120. doi: 10.3934/era.2013.20.109
[9]

Aravind Asok, James Parson. Equivariant sheaves on some spherical varieties. Electronic Research Announcements, 2011, 18: 119-130. doi: 10.3934/era.2011.18.119
[10]

Xiao-Qiang Zhao, Shengfan Zhou. Kernel sections for processes and nonautonomous lattice systems. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 763-785. doi: 10.3934/dcdsb.2008.9.763
[11]

Michael Gekhtman, Michael Shapiro, Serge Tabachnikov, Alek Vainshtein. Higher pentagram maps, weighted directed networks, and cluster dynamics. Electronic Research Announcements, 2012, 19: 1-17. doi: 10.3934/era.2012.19.1
[12]

Gurkan Ozturk, Mehmet Tahir Ciftci. Clustering based polyhedral conic functions algorithm in classification. Journal of Industrial & Management Optimization, 2015, 11 (3) : 921-932. doi: 10.3934/jimo.2015.11.921
[13]

Jesús Carrillo-Pacheco, Felipe Zaldivar. On codes over FFN$(1,q)$-projective varieties. Advances in Mathematics of Communications, 2016, 10 (2) : 209-220. doi: 10.3934/amc.2016001
[14]

Sylvain E. Cappell, Anatoly Libgober, Laurentiu Maxim and Julius L. Shaneson. Hodge genera and characteristic classes of complex algebraic varieties. Electronic Research Announcements, 2008, 15: 1-7. doi: 10.3934/era.2008.15.1
[15]

Alfonso Artigue. Discrete and continuous topological dynamics: Fields of cross sections and expansive flows. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 5911-5927. doi: 10.3934/dcds.2016059
[16]

Shengfan Zhou, Linshan Wang. Kernel sections for damped non-autonomous wave equations with critical exponent. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 399-412. doi: 10.3934/dcds.2003.9.399
[17]

Michal Kočvara, Jiří V. Outrata. Inverse truss design as a conic mathematical program with equilibrium constraints. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1329-1350. doi: 10.3934/dcdss.2017071
[18]

Lijuan Zhao, Wenyu Sun. Nonmonotone retrospective conic trust region method for unconstrained optimization. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 309-325. doi: 10.3934/naco.2013.3.309
[19]

Jing Zhou, Dejun Chen, Zhenbo Wang, Wenxun Xing. A conic approximation method for the 0-1 quadratic knapsack problem. Journal of Industrial & Management Optimization, 2013, 9 (3) : 531-547. doi: 10.3934/jimo.2013.9.531
[20]

Xin Zhang, Jie Wen, Qin Ni. Subspace trust-region algorithm with conic model for unconstrained optimization. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 223-234. doi: 10.3934/naco.2013.3.223

2018 Impact Factor: 0.263

Tools

Metrics

  • PDF downloads (44)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences