Schottky algorithms: Classical meets tropical

Chua, Lynn; Kummer, Mario; Sturmfels, Bernd

doi:10.1090/mcom/3406

Schottky algorithms: Classical meets tropical
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Math. Comp. 88 (2019), 2541-2558 Request permission

Abstract:

We present a new perspective on the Schottky problem that links numerical computing with tropical geometry. The task is to decide whether a symmetric matrix defines a Jacobian, and, if so, to compute the curve and its canonical embedding. We offer solutions and their implementations in genus four, both classically and tropically. The locus of cographic matroids arises from tropicalizing the Schottky–Igusa modular form.

References

Barbara Bolognese, Madeline Brandt, and Lynn Chua, From curves to tropical Jacobians and back, Combinatorial algebraic geometry, Fields Inst. Commun., vol. 80, Fields Inst. Res. Math. Sci., Toronto, ON, 2017, pp. 21–45. MR 3752493
Silvia Brannetti, Margarida Melo, and Filippo Viviani, On the tropical Torelli map, Adv. Math. 226 (2011), no. 3, 2546–2586. MR 2739784, DOI 10.1016/j.aim.2010.09.011
Melody Chan, Combinatorics of the tropical Torelli map, Algebra Number Theory 6 (2012), no. 6, 1133–1169. MR 2968636, DOI 10.2140/ant.2012.6.1133
Francesco Dalla Piazza, Alessio Fiorentino, and Riccardo Salvati Manni, Plane quartics: the universal matrix of bitangents, Israel J. Math. 217 (2017), no. 1, 111–138. MR 3625106, DOI 10.1007/s11856-017-1440-z
Bernard Deconinck, Matthias Heil, Alexander Bobenko, Mark van Hoeij, and Marcus Schmies, Computing Riemann theta functions, Math. Comp. 73 (2004), no. 247, 1417–1442. MR 2047094, DOI 10.1090/S0025-5718-03-01609-0
Bernard Deconinck and Mark van Hoeij, Computing Riemann matrices of algebraic curves, Phys. D 152/153 (2001), 28–46. Advances in nonlinear mathematics and science. MR 1837895, DOI 10.1016/S0167-2789(01)00156-7
M. Dutour Sikirić, Polyhedral, a GAP package, mathieudutour.altervista.org/Polyhedral, 2013.
Mathieu Dutour Sikirić, Alexey Garber, Achill Schürmann, and Clara Waldmann, The complete classification of five-dimensional Dirichlet-Voronoi polyhedra of translational lattices, Acta Crystallogr. Sect. A 72 (2016), no. 6, 673–683. MR 3573502, DOI 10.1107/s2053273316011682
Ewgenij Gawrilow and Michael Joswig, polymake: a framework for analyzing convex polytopes, Polytopes—combinatorics and computation (Oberwolfach, 1997) DMV Sem., vol. 29, Birkhäuser, Basel, 2000, pp. 43–73. MR 1785292
Alain Ghouila-Houri, Caractérisation des matrices totalement unimodulaires, C. R. Acad. Sci. Paris 254 (1962), 1192–1194 (French). MR 132752
Samuel Grushevsky, The Schottky problem, Current developments in algebraic geometry, Math. Sci. Res. Inst. Publ., vol. 59, Cambridge Univ. Press, Cambridge, 2012, pp. 129–164. MR 2931868
Samuel Grushevsky and Martin Möller, Explicit formulas for infinitely many Shimura curves in genus 4, Asian J. Math. 22 (2018), no. 2, 381–390. MR 3824574, DOI 10.4310/ajm.2018.v22.n2.a12
Christian Haase, Gregg Musiker, and Josephine Yu, Linear systems on tropical curves, Math. Z. 270 (2012), no. 3-4, 1111–1140. MR 2892941, DOI 10.1007/s00209-011-0844-4
Jonathan D. Hauenstein and Andrew J. Sommese, What is numerical algebraic geometry? [Foreword]. part 3, J. Symbolic Comput. 79 (2017), no. part 3, 499–507. MR 3563094, DOI 10.1016/j.jsc.2016.07.015
Jun-ichi Igusa, On the irreducibility of Schottky’s divisor, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 531–545 (1982). MR 656035
E. Jones et al., SciPy: Open source scientific tools for Python, 2001-, http://www.scipy.org/.
George R. Kempf, The equations defining a curve of genus $4$, Proc. Amer. Math. Soc. 97 (1986), no. 2, 219–225. MR 835869, DOI 10.1090/S0002-9939-1986-0835869-2
Bo Lin, Computing linear systems on metric graphs, J. Symbolic Comput. 87 (2018), 54–67. MR 3744339, DOI 10.1016/j.jsc.2017.05.007
Grigory Mikhalkin and Ilia Zharkov, Tropical curves, their Jacobians and theta functions, Curves and abelian varieties, Contemp. Math., vol. 465, Amer. Math. Soc., Providence, RI, 2008, pp. 203–230. MR 2457739, DOI 10.1090/conm/465/09104
John Christian Ottem, Kristian Ranestad, Bernd Sturmfels, and Cynthia Vinzant, Quartic spectrahedra, Math. Program. 151 (2015), no. 2, Ser. B, 585–612. MR 3348164, DOI 10.1007/s10107-014-0844-3
W. Plesken and B. Souvignier, isom and autom, 1995, published under GPL licence at http://www.math.uni-rostock.de/$\sim$waldmann/ISOM_and_AUTO.zip
Qingchun Ren, Steven V. Sam, Gus Schrader, and Bernd Sturmfels, The universal Kummer threefold, Exp. Math. 22 (2013), no. 3, 327–362. MR 3171096, DOI 10.1080/10586458.2013.816206
F. Schottky, Zur Theorie der Abelschen Functionen von vier Variabeln, J. Reine Angew. Math. 102 (1888), 304–352 (German). MR 1580147, DOI 10.1515/crll.1888.102.304
Christopher Swierczewski and Bernard Deconinck, Computing Riemann theta functions in Sage with applications, Math. Comput. Simulation 127 (2016), 263–272. MR 3501304, DOI 10.1016/j.matcom.2013.04.018
C. Swierczewski et al., Abelfunctions: A library for computing with Abelian functions, Riemann surfaces, and algebraic curves, github.com/abelfunctions/abelfunctions, 2016.
W. T. Tutte, An algorithm for determining whether a given binary matroid is graphic, Proc. Amer. Math. Soc. 11 (1960), 905–917. MR 117173, DOI 10.1090/S0002-9939-1960-0117173-5
F. Vallentin, Sphere covering, lattices, and tilings (in low dimensions), PhD thesis, TU München, 2003.
Georges Voronoi, Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Deuxième mémoire. Recherches sur les parallélloèdres primitifs, J. Reine Angew. Math. 134 (1908), 198–287 (French). MR 1580754, DOI 10.1515/crll.1908.134.198