Stable intersections of tropical varieties. (English) Zbl 1406.14045
Summary: We give several characterizations of stable intersections of tropical cycles and establish their fundamental properties. We prove that the stable intersection of two tropical varieties is the tropicalization of the intersection of the classical varieties after a generic rescaling. A proof of Bernstein’s theorem follows from this. We prove that the tropical intersection ring of tropical cycle fans is isomorphic to McMullen’s polytope algebra. It follows that every tropical cycle fan is a linear combination of pure powers of tropical hypersurfaces, which are always realizable. We prove that every stable intersection of constant coefficient tropical varieties defined by prime ideals is connected through codimension one. We also give an example of a realizable tropical variety that is connected through codimension one but whose stable intersection with a hyperplane is not.
MSC:
| 14T05 | Tropical geometry (MSC2010) |
| 14C17 | Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry |
References:
| [1] | Allermann, L., Rau, J.: First steps in tropical intersection theory. Math. Z. 264(3), 633-670 (2010) · Zbl 1193.14074 · doi:10.1007/s00209-009-0483-1 |
| [2] | Babaee, F., Huh, J.: A tropical approach to the strongly positive Hodge conjecture. arXiv:1502.00299 · Zbl 1396.14064 |
| [3] | Bogart, T., Jensen, A.N., Speyer, D., Sturmfels, B., Thomas, R.R.: Computing tropical varieties. J. Symb. Comput. 42(1-2), 54-73 (2007) · Zbl 1121.14051 · doi:10.1016/j.jsc.2006.02.004 |
| [4] | Cartwright, D., Payne, S.: Connectivity of tropicalizations. Math. Res. Lett. 19(5), 1089-1095 (2012) · Zbl 1291.14091 · doi:10.4310/MRL.2012.v19.n5.a10 |
| [5] | Fulton, W., Sturmfels, B.: Intersection theory on toric varieties. Topology 36(2), 335-353 (1997) · Zbl 0885.14025 · doi:10.1016/0040-9383(96)00016-X |
| [6] | Hampe, S.: a-tint: a polymake extension for algorithmic tropical intersection theory. Eur. J. Combin. 36, 579-607 (2014) · Zbl 1285.14071 · doi:10.1016/j.ejc.2013.10.001 |
| [7] | Jensen, A.N.: Gfan, a software system for Gröbner fans and tropical varieties. http://home.imf.au.dk/jensen/software/gfan/gfan.html · Zbl 1291.14091 |
| [8] | Jensen, A., Yu, J.: Computing tropical resultants. J. Algebra 387, 287-319 (2013) · Zbl 1316.14117 · doi:10.1016/j.jalgebra.2013.03.031 |
| [9] | Katz, E.: Tropical intersection theory from toric varieties. Collect. Math. 63(1), 29-44 (2012) · Zbl 1322.14091 · doi:10.1007/s13348-010-0014-8 |
| [10] | Kazarnovskiĭ, B.Y.: c-fans and Newton polyhedra of algebraic varieties. Izv. Ross. Akad. Nauk Ser. Mat. 67(3), 23-44 (2003) · Zbl 1077.14072 · doi:10.4213/im434 |
| [11] | McMullen, P.: The polytope algebra. Adv. Math. 78(1), 76-130 (1989) · Zbl 0686.52005 · doi:10.1016/0001-8708(89)90029-7 |
| [12] | McMullen, P.: On simple polytopes. Invent. Math. 113(2), 419-444 (1993) · Zbl 0803.52007 · doi:10.1007/BF01244313 |
| [13] | Mikhalkin, G.: Tropical geometry and its applications. In: Proceedings of the International Congress of Mathematicians, vol. II, pp. 827-852 (Madrid, 2006, Zürich, Switzerland). European Mathematical Society (2006) · Zbl 1103.14034 |
| [14] | Maclagan, D., Sturmfels, B.: Introduction to Tropical Geometry, Graduate Studies in Mathematics, vol. 161. American Mathematical Society, Providence (2015) · Zbl 1321.14048 |
| [15] | Osserman, B., Payne, S.: Lifting tropical intersections. Doc. Math. 18, 121-175 (2013) · Zbl 1308.14069 |
| [16] | Rau, J.: Intersections on tropical moduli spaces. arXiv:0812.3678 · Zbl 1379.14035 |
| [17] | Richter-Gebert, J., Sturmfels, B., Theobald, T.: First steps in tropical geometry. In: Idempotent Mathematics and Mathematical Physics, Contemporary Mathematics, vol. 377, pp. 289-317. American Mathematical Society, Providence (2005) · Zbl 1093.14080 |
| [18] | Sturmfels, B., Tevelev, J.: Elimination theory for tropical varieties. Math. Res. Lett. 15(3), 543-562 (2008) · Zbl 1157.14038 · doi:10.4310/MRL.2008.v15.n3.a14 |
| [19] | Sturmfels, B.: Solving systems of polynomial equations. In: CBMS Regional Conference Series in Mathematics, vol. 97. Published for the Conference Board of the Mathematical Sciences, Washington (2002) · Zbl 1101.13040 |
| [20] | Tverberg, H.: How to cut a convex polytope into simplices. Geom. Dedicata 3, 239-240 (1974) · Zbl 0288.52006 · doi:10.1007/BF00183215 |
| [21] | Yu, J.: Algebraic matroids and realizability of tropical varieties up to scaling. arXiv:1506.01427 · Zbl 1308.14069 |
| [22] | Ziegler, G.M.: Lectures on Polytopes, Graduate Texts in Mathematics, vol. 152. Springer, New York (1995) · Zbl 0823.52002 · doi:10.1007/978-1-4613-8431-1 |
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