Abstract
K3 polytopes appear in complements of tropical quartic surfaces. They are dual to regular unimodular central triangulations of reflexive polytopes in the fourth dilation of the standard tetrahedron. Exploring these combinatorial objects, we classify K3 polytopes with up to 30 vertices. Their number is 36 297 333. We study the singular loci of quartic surfaces that tropicalize to K3 polytopes. These surfaces are stable in the sense of Geometric Invariant Theory.
Funding statement: GB was partially supported by the Vetenskapsrådet grant NT:2014-3991. MP and BS acknowledge support by the Einstein Foundation Berlin, which also funded a visit of GB to TU Berlin.
Acknowledgements
We are very grateful to Michael Joswig for several inspiring discussions.We also thank Matteo Gallet, Lars Kastner and Benjamin Schröter for help with this project.
Communicated by: R. Cavalieri
References
[1] V. I. Arnol’d, Normal forms of functions near degenerate critical points, the Weyl groups Ak , Dk , Ek and Lagrangian singularities. Funkcional. Anal. i Priložen. 6 (1972), 3–25. English translation: Functional Anal. Appl. 6 (1972), 254–272. MR0356124 Zbl 0278.5701110.1007/BF01077644Search in Google Scholar
[2] V. I. Arnol’d, Normal forms of functions in the neighborhood of degenerate critical points. Uspehi Mat. Nauk 29 (1974), 11–49. English translation: Russian Math. Surveys 29 (1974), no. 2, 10–50. MR0516034 Zbl 0298.57022Search in Google Scholar
[3] V. V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties. J. Algebraic Geom. 3 (1994), 493–535. MR1269718 Zbl 0829.14023Search in Google Scholar
[4] J. A. De Loera, J. Rambau, F. Santos, Triangulations volume 25 of Algorithms and Computation in Mathematics Springer 2010. MR2743368 Zbl 1207.5200210.1007/978-3-642-12971-1Search in Google Scholar
[5] M. Develin, B. Sturmfels, Tropical convexity. Doc. Math. 9 (2004), 1–27. MR2054977 Zbl 1054.5200410.4171/dm/154Search in Google Scholar
[6] E. Gawrilow, M. Joswig, polymake: a framework for analyzing convex polytopes. In: Polytopes—combinatorics and computationOberwolfach, 1997), volume 29 of DMV Sem. 43–73, Birkhäuser, Basel 2000. MR1785292 Zbl 0960.6818210.1007/978-3-0348-8438-9_2Search in Google Scholar
[7] G.-M. Greuel, C. Lossen, E. Shustin, Introduction to singularities and deformations Springer 2007. MR2290112 Zbl 1125.32013Search in Google Scholar
[8] T. Hibi, Dual polytopes of rational convex polytopes. Combinatorica 12 (1992), 237–240. MR1179260 Zbl 0758.5200910.1007/BF01204726Search in Google Scholar
[9] A. N. Jensen, Gfan, a software system for Gröbner fans and tropical varietieshttp://home.imf.au.dk/jensen/software/gfan/gfan.htmlSearch in Google Scholar
[10] C. Jordan, M. Joswig, L. Kastner, Parallel enumeration of triangulations. Electron. J. Combin. 25 (2018), Paper 3.6, 27 pp. MR3829292 Zbl 1393.6817510.37236/7318Search in Google Scholar
[11] M. Joswig, K. Kulas, Tropical and ordinary convexity combined. Adv. Geom. 10 (2010), 333–352. MR2629819 Zbl 1198.1406010.1515/advgeom.2010.012Search in Google Scholar
[12] M. Joswig, G. Loho, Weighted digraphs and tropical cones. Linear Algebra Appl. 501 (2016), 304–343. MR3485070 Zbl 1405.1414110.1016/j.laa.2016.02.027Search in Google Scholar
[13] M. Joswig, B. Sturmfels, J. Yu, Affine buildings and tropical convexity. Albanian J. Math. 1 (2007), 187–211. MR2367213 Zbl 1133.52003Search in Google Scholar
[14] A. M. Kasprzyk, Canonical toric Fano threefolds. Canad. J. Math. 62 (2010), 1293–1309. MR2760660 Zbl 1264.1405510.4153/CJM-2010-070-3Search in Google Scholar
[15] E. Katz, H. Markwig, T. Markwig, The j-invariant of a plane tropical cubic. J. Algebra 320 (2008), 3832–3848. MR2457725 Zbl 1185.1403010.1016/j.jalgebra.2008.08.018Search in Google Scholar
[16] M. Kreuzer, H. Skarke, Classification of reflexive polyhedra in three dimensions. Adv. Theor. Math. Phys. 2 (1998), 853–871. MR1663339 Zbl 0934.5200610.4310/ATMP.1998.v2.n4.a5Search in Google Scholar
[17] M. Kreuzer, H. Skarke, Complete classification of reflexive polyhedra in four dimensions. Adv. Theor. Math. Phys. 4 (2000), 1209–1230. MR1894855 Zbl 1017.5200710.4310/ATMP.2000.v4.n6.a2Search in Google Scholar
[18] T. Lam, A. Postnikov, Alcoved polytopes. I. Discrete Comput. Geom. 38 (2007), 453–478. MR2352704 Zbl 1134.5201910.1007/s00454-006-1294-3Search in Google Scholar
[19] D. Maclagan, B. Sturmfels, Introduction to tropical geometry volume 161 of Graduate Studies in Mathematics Amer. Math. Soc. 2015. MR3287221 Zbl 1321.1404810.1090/gsm/161Search in Google Scholar
[20] D. Mumford, Stability of projective varieties L’Enseignement Mathématique, Geneva 1977. MR0450273 Zbl 0376.14007Search in Google Scholar
[21] D. Mumford, J. Fogarty, F. Kirwan, Geometric invariant theory Springer 1994. MR1304906 Zbl 0797.1400410.1007/978-3-642-57916-5Search in Google Scholar
[22] J. Rambau, TOPCOM: triangulations of point configurations and oriented matroids. In: Mathematical softwareBeijing, 2002), 330–340, World Sci. Publ., River Edge, NJ 2002. MR1932619 Zbl 1057.6815010.1142/9789812777171_0035Search in Google Scholar
[23] J. Shah, Degenerations of K3 surfaces of degree 4. Trans. Amer. Math. Soc. 263 (1981), 271–308. MR594410 Zbl 0456.1401910.2307/1998352Search in Google Scholar
[24] N. M. Tran, Enumerating polytropes. J. Combin. Theory Ser. A 151 (2017), 1–22. MR3663485 Zbl 0674486410.1016/j.jcta.2017.03.011Search in Google Scholar
[25] M. Vigeland, The group law on a tropical elliptic curve. Math. Scand. 104 (2009), 188–204. Zbl 1167.1401810.7146/math.scand.a-15094Search in Google Scholar
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