# Stable intersections of tropical varieties

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## Abstract

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.

## Keywords

Tropical geometry Intersection theory Polytope algebra## Notes

### Acknowledgments

We thank Diane Maclagan for reading and providing feedback on an earlier draft and the referees for many helpful suggestions that greatly improved the exposition. The first author was supported by the Danish Council for Independent Research, Natural Sciences (FNU), and the second author was supported by the NSF Grant DMS #1101289.

## References

- 1.Allermann, L., Rau, J.: First steps in tropical intersection theory. Math. Z.
**264**(3), 633–670 (2010)MATHMathSciNetCrossRefGoogle Scholar - 2.Babaee, F., Huh, J.: A tropical approach to the strongly positive Hodge conjecture. arXiv:1502.00299
- 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)MATHMathSciNetCrossRefGoogle Scholar - 4.Cartwright, D., Payne, S.: Connectivity of tropicalizations. Math. Res. Lett.
**19**(5), 1089–1095 (2012)MATHMathSciNetCrossRefGoogle Scholar - 5.Fulton, W., Sturmfels, B.: Intersection theory on toric varieties. Topology
**36**(2), 335–353 (1997)MATHMathSciNetCrossRefGoogle Scholar - 6.Hampe, S.: a-tint: a polymake extension for algorithmic tropical intersection theory. Eur. J. Combin.
**36**, 579–607 (2014)MATHMathSciNetCrossRefGoogle Scholar - 7.Jensen, A.N.: Gfan, a software system for Gröbner fans and tropical varieties. http://home.imf.au.dk/jensen/software/gfan/gfan.html
- 8.Jensen, A., Yu, J.: Computing tropical resultants. J. Algebra
**387**, 287–319 (2013)MATHMathSciNetCrossRefGoogle Scholar - 9.Katz, E.: Tropical intersection theory from toric varieties. Collect. Math.
**63**(1), 29–44 (2012)MATHMathSciNetCrossRefGoogle Scholar - 10.Kazarnovskiĭ, B.Y.: c-fans and Newton polyhedra of algebraic varieties. Izv. Ross. Akad. Nauk Ser. Mat.
**67**(3), 23–44 (2003)MathSciNetCrossRefGoogle Scholar - 11.McMullen, P.: The polytope algebra. Adv. Math.
**78**(1), 76–130 (1989)MATHMathSciNetCrossRefGoogle Scholar - 12.McMullen, P.: On simple polytopes. Invent. Math.
**113**(2), 419–444 (1993)MATHMathSciNetCrossRefGoogle Scholar - 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)Google Scholar
- 14.Maclagan, D., Sturmfels, B.: Introduction to Tropical Geometry, Graduate Studies in Mathematics, vol. 161. American Mathematical Society, Providence (2015)Google Scholar
- 15.Osserman, B., Payne, S.: Lifting tropical intersections. Doc. Math.
**18**, 121–175 (2013)MATHMathSciNetGoogle Scholar - 16.Rau, J.: Intersections on tropical moduli spaces. arXiv:0812.3678
- 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)Google Scholar
- 18.Sturmfels, B., Tevelev, J.: Elimination theory for tropical varieties. Math. Res. Lett.
**15**(3), 543–562 (2008)MATHMathSciNetCrossRefGoogle Scholar - 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)Google Scholar
- 20.Tverberg, H.: How to cut a convex polytope into simplices. Geom. Dedicata
**3**, 239–240 (1974)MATHMathSciNetCrossRefGoogle Scholar - 21.Yu, J.: Algebraic matroids and realizability of tropical varieties up to scaling. arXiv:1506.01427
- 22.Ziegler, G.M.: Lectures on Polytopes, Graduate Texts in Mathematics, vol. 152. Springer, New York (1995)CrossRefGoogle Scholar