×

zbMATH — the first resource for mathematics

Lifting tropical bitangents. (English) Zbl 07074721
Summary: We study lifts of tropical bitangents to the tropicalization of a given complex algebraic curve together with their lifting multiplicities. Using this characterization, we show that generically all the seven bitangents of a smooth tropical plane quartic lift in sets of four to algebraic bitangents. We do this constructively, i.e. we give solutions for the initial terms of the coefficients of the bitangent lines. This is a step towards a tropical proof that a general smooth quartic admits 28 bitangent lines. The methods are also appropriate to count real bitangents; however the conditions to determine whether a tropical bitangent has real lifts are not purely combinatorial.

MSC:
14T05 Tropical geometry (MSC2010)
68W30 Symbolic computation and algebraic computation
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Allamigeon, X.; Benchimol, P.; Gaubert, S.; Joswig, M., Tropicalizing the simplex algorithm, SIAM J. Discrete Math., 29, 2, 751-795, (2015) · Zbl 1334.14033
[2] Allermann, L.; Rau, J., First steps in tropical intersection theory, Math. Z., 264, 3, 633-670, (2010) · Zbl 1193.14074
[3] Amini, O.; Baker, M.; Brugallé, E.; Rabinoff, J., Lifting harmonic morphisms I: metrized complexes and Berkovich skeleta, Res. Math. Sci., 2, Article 7 pp., (2015) · Zbl 1327.14117
[4] Amini, O.; Baker, M.; Brugallé, E.; Rabinoff, J., Lifting harmonic morphisms II: tropical curves and metrized complexes, Algebra Number Theory, 9, 2, 267-315, (2015) · Zbl 1312.14138
[5] Baker, M.; Jensen, D., Degeneration of Linear Series from the Tropical Point of View and Applications, 365-433, (2016), Springer International Publishing: Springer International Publishing Cham · Zbl 1349.14193
[6] Baker, M.; Len, Y.; Morrison, R.; Pflueger, N.; Ren, Q., Bitangents of tropical plane quartic curves, Math. Z., 1-15, (2015)
[7] Brugallé, E.; López de Medrano, Lucia M., Inflection points of real and tropical plane curves, J. Singul., 4, 74-103, (2012) · Zbl 1292.14042
[8] Chan, M.; Jiradilok, P., Theta characteristics of tropical \(k_4\)-curves, (2015), Preprint · Zbl 1390.14195
[9] Cueto, María Angélica; Markwig, Hannah, How to repair tropicalizations of plane curves using modifications, Exp. Math., 25, 2, 130-164, (2016) · Zbl 1349.14196
[10] Decker, Wolfram; Greuel, Gert-Martin; Pfister, Gerhard; Schönemann, Hans, Singular 3-1-3. A Computer Algebra System for Polynomial Computations, (2011), Centre for Computer Algebra, University of Kaiserslautern
[11] Einsiedler, Manfred; Kapranov, Mikhail; Lind, Douglas, Non-archimedean amoebas and tropical varieties, J. Reine Angew. Math., 601, 139-157, (2006) · Zbl 1115.14051
[12] Eisenbud, D., Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, (1995), Springer · Zbl 0819.13001
[13] Eisenbud, D.; Harris, J., 3264 and All That, (2016), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 1341.14001
[14] Gathmann, A., Tropical algebraic geometry, Jahresber. Dtsch. Math.-Ver., 108, 1, 3-32, (2006) · Zbl 1109.14038
[15] Insong, C., Multi-tangent space for algebraic curves. MathOverflow
[16] Itenberg, Ilia; Mikhalkin, Grigory; Shustin, Eugenii, Tropical Algebraic Geometry, Oberwolfach Seminars, vol. 35, (2009), Birkhäuser Verlag: Birkhäuser Verlag Basel · Zbl 1165.14002
[17] Jensen, D.; Len, Y., Tropicalization of theta characteristics, double covers, and Prym varieties, (June 2016), Preprint
[18] Jensen, Christian U.; Lenzing, Helmut, Model-Theoretic Algebra with Particular Emphasis on Fields, Rings, Modules, Algebra, Logic and Applications, vol. 2, (1989), Gordon and Breach Science Publishers: Gordon and Breach Science Publishers New York · Zbl 0728.03026
[19] Jensen, Anders N.; Markwig, Hannah; Markwig, Thomas, tropical.lib. A Singular 3.0 library for computations in tropical geometry, (2007)
[20] Kajiwara, Takeshi, Tropical toric geometry, (Toric Topology. Toric Topology, Contemp. Math., vol. 460, (2008), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 197-207 · Zbl 1202.14047
[21] Markwig, Hannah; Markwig, Thomas; Shustin, Eugenii, Tropical curves with a singularity in a fixed point, Manuscr. Math., (2011) · Zbl 1254.14070
[22] Markwig, Hannah; Markwig, Thomas; Shustin, Eugenii, Enumeration of complex and real surfaces via tropical geometry, (2015), Preprint · Zbl 1408.14203
[23] Mikhalkin, Grigory, Tropical geometry and its applications, (Sanz-Sole, M.; etal., Invited Lectures v. II. Invited Lectures v. II, Proceedings of the ICM Madrid, (2006)), 827-852 · Zbl 1103.14034
[24] Morrison, Ralph, Tropical images of intersection points, Collect. Math., 66, 2, 273-283, (2015) · Zbl 1331.14059
[25] O’Neill, C., Owusu Kwaakwah, E., de Wolff, T., 2015 - present. Viro.sage. A Python class for using Viro Patchworking; Version 0.3a.
[26] Osserman, B.; Payne, S., Lifting tropical intersections, Doc. Math., 18, 121-175, (2013) · Zbl 1308.14069
[27] Osserman, B.; Rabinoff, J., Lifting nonproper tropical intersections, (Tropical and Non-Archimedean Geometry. Tropical and Non-Archimedean Geometry, Contemp. Math., vol. 605, (2013), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 15-44 · Zbl 1320.14078
[28] Panizzut, M., Theta characteristics of hyperelliptic graphs, (2015), Preprint · Zbl 1375.14217
[29] Payne, S., Analytification is the limit of all tropicalizations, Math. Res. Lett., 16, 3, 543-556, (2009) · Zbl 1193.14077
[30] Plaumann, D.; Sturmfels, B.; Vinzant, C., Quartic curves and their bitangents, J. Symb. Comput., 46, 712-733, (2011) · Zbl 1214.14049
[31] Plücker, J., Solution d’une question fondamentale concernant la théorie générale des courbes, J. Reine Angew. Math., 12, 105-108, (1834)
[32] Richter-Gebert, J.; Sturmfels, B.; Theobald, T., First steps in tropical geometry, Contemp. Math., 377, 289-317, (2005) · Zbl 1093.14080
[33] Shaw, K., A tropical intersection product in matroidal fans, SIAM J. Discrete Math., 27, 1, 459-491, (2013) · Zbl 1314.14113
[34] Tiwari, A.; Zheng, W., A bitangent class that lifts more than 4 times
[35] Viro, O. Ya., Real plane algebraic curves: constructions with controlled topology, Algebra Anal., 1, 5, 1-73, (1989)
[36] Zharkov, I., Tropical theta characteristics, (Mirror Symmetry and Tropical Geometry. Mirror Symmetry and Tropical Geometry, Contemp. Math., vol. 527, (2010), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 165-168 · Zbl 1213.14120
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.