Higher balancing for tropical manifolds. (English) Zbl 1532.14106
The authors generalize the well-known balancing condition of Bergman Fans to cones of higher codimension. Since Bergman fans are the local patches of tropical manifolds, these new balancing results hold for all tropical manifolds. The authors establish a reinterpretation of the balancing of the Chern-Schwartz-MacPherson cycle, which was introduced by L. López de Medrano et al. [Proc. Lond. Math. Soc. (3) 120, No. 1, 1–27 (2020; Zbl 1454.14013)].
Reviewer: Kevin Kühn (Frankfurt am Main)
Citations:
Zbl 1454.14013References:
| [1] | Mikhalkin, G., Zharkov, I.: Tropical eigenwave and intermediate Jacobians. In: Homological Mirror Symmetry and Tropical Geometry, volume 15 of Lect. Notes Unione Mat. Ital., pp 309-349. Springer, Cham, (2014) · Zbl 1408.14204 |
| [2] | Shaw, K.: Tropical surfaces. Preprint, arxiv:1506.07407, (2015) |
| [3] | Hampe, S., Tropical linear spaces and tropical convexity, Electron. J. Combin., 22, 4, 20 (2015) · Zbl 1330.14099 · doi:10.37236/5271 |
| [4] | Speyer, DE, Tropical linear spaces, SIAM J. Discret. Math., 22, 4, 1527-1558 (2008) · Zbl 1191.14076 · doi:10.1137/080716219 |
| [5] | Lorscheid, O.: A unifying approach to tropicalization. To appear in Transactions of the American Mathematical Society · Zbl 07678797 |
| [6] | Izhakian, Zur; Shustin, Eugenii, Idempotent semigroups and tropical algebraic sets, J. Eur. Math. Soc. (JEMS), 14, 2, 489-520 (2012) · Zbl 1273.14133 |
| [7] | Maclagan, Diane; Rincón, Felipe, Tropical ideals, Compos. Math., 154, 3, 640-670 (2018) · Zbl 1428.14093 · doi:10.1112/S0010437X17008004 |
| [8] | Maclagan, D., Rincón, F.: Varieties of tropical ideals are balanced. Preprint, arxiv:2009.14557, (2020) |
| [9] | Giansiracusa, J.; Giansiracusa, N., Equations of tropical varieties, Duke Math. J., 165, 18, 3379-3433 (2016) · Zbl 1409.14100 · doi:10.1215/00127094-3645544 |
| [10] | Maclagan, D.; Rincón, F., Tropical schemes, tropical cycles, and valuated matroids, J. Eur. Math. Soc. (JEMS), 22, 3, 777-796 (2020) · Zbl 1509.14120 · doi:10.4171/JEMS/932 |
| [11] | Bernd, S.: Solving systems of polynomial equations. In: CBMS Regional Conference Series in Mathematics, vol. 97. American Mathematical Society, Providence, RI (2002) · Zbl 1101.13040 |
| [12] | Ardila, F.; Klivans, CJ, The Bergman complex of a matroid and phylogenetic trees, J. Combin. Theory Ser. B, 96, 1, 38-49 (2006) · Zbl 1082.05021 · doi:10.1016/j.jctb.2005.06.004 |
| [13] | Huh, J.: Rota’s conjecture and positivity of algebraic cycles in permutohedral varieties. Thesis. Online available at http://www-personal.umich.edu/ junehuh/thesis.pdf, (2014) |
| [14] | William, F.: Introduction to toric varieties. In: volume 131 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ. The William H. Roever Lectures in Geometry (1993) · Zbl 0813.14039 |
| [15] | Speyer, D. E.: Tropical geometry. Thesis. Online available at http://www-personal.umich.edu/ speyer/thesis.pdf, (2005) |
| [16] | Oxley, J.: Matroid Theory. Second edn. Oxford Graduate Texts in Mathematics, vol. 21. Oxford University Press, Oxford, 2011. xiv+684 pp. ISBN: 978-0-19-960339-8 · Zbl 1254.05002 |
| [17] | Medrano, Lucía López de; Rincón, Felipe; Shaw, Kris, Chern-Schwartz-MacPherson cycles of matroids, Proc. Lond. Math. Soc. (3), 120, 1, 1-27 (2020) · Zbl 1454.14013 · doi:10.1112/plms.12278 |
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