Abstract
This article studies the relationship between tropical Severi varieties and secondary fans. In the case when tropical Severi varieties are hypersurfaces this relationship is very well known; specifically, in this case, a tropical Severi variety of codimension 1 is a subfan of the corresponding secondary fan. It was expected for some time that this continues to hold more generally, but Katz found a counterexample in codimension 2, showing that this relationship is more subtle. The two main results in this paper are as follows. The first theorem finds a simple condition under which a tropical Severi variety cannot be a subfan of the corresponding secondary fan. The second theorem provides a partial converse, namely, we find conditions under which a cone of the secondary fan is fully contained in the tropical Severi variety. As a first application of these results, we also find a combinatorial formula for the tropical intersection multiplicities for secondary fans.
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References
Allermann L., Rau J.: First steps in tropical intersection theory. Math. Z. 264(3), 633–670 (2010)
Bieri R., Groves J.: The geometry of the set of characters induced by valuations. J. Reine Angew. Math. 347, 168–195 (1984)
Caporaso L., Harris J.: Counting plane curves of any genus. Invent. Math. 131(2), 345–392 (1998)
Cox, D., Little, J., Schenck, H.: Toric Varieties, Graduate Studies in Mathematics, vol. 124. AMS, Providence, RI (2011)
Dickenstein A., Feichtner E.M., Sturmfels B.: Tropical discriminants. J. Am. Math. Soc. 20, 1111–1133 (2007)
Enriques, F.: Sui moduli d’una classe di superficie e sul teorema d’esistenza per funzioni algebriche di due variabilis. Atti Accad. Sci. Torino. 47, 300–307 (1912).
Fulton, W.: On nodal curves. In: Algebraic Geometry – Open Problems. Proceedings of the Conference Held in Ravello, May 31 – June 5, 1982. Lecture Notes in Mathematics, vol. 997, pp. 146–155. Springer, Berlin, Heidelberg (1983)
Gelfand I.M., Kapranov M.M., Zelevinsky A.V.: Discriminants, Resultants, and Multidimensional Determinants. Birkhauser, Boston (1994)
Itenberg I., Mikhalkin G., Shustin E.: Tropical Algebraic Geometry. Oberwolfach Seminars, vol. 35. Birkhauser, Basel (2007)
Katz E.: Tropical invariants from the secondary fan. Adv. Geom. 9(2), 153–180 (2009)
Katz E.: A tropical toolkit. Expos. Math. 27(1), 1–36 (2009)
Kazarnovskii, B.Y.: c-Fans and Newton polyhedra of algebraic varieties. Izv. Ross. Akad. Nauk Ser. Math. 67(3), 23–44 (2003) (Russian); (Translation in Izv. Math. 67(3), 439–460 (2003)).
Maclagan, D., Sturmfels, B.: Introduction to Tropical Geometry. Graduate Studies in Mathematics, vol. 161. AMS, Providence, RI (2015)
Markwig H., Markwig T., Shustin E.: Tropical curves with a singularity in a fixed point. Manuscr. Math. 137, 383–418 (2012)
Mikhalkin G.: Enumerative tropical algebraic geometry in \({\mathbb{R}^{2}}\). J. Am. Math. Soc. 18, 313–377 (2005)
Severi F.: Vorlesungen über Algebraische Geometrie. Teubner, Leipzig (1921)
Shustin, E.: A tropical approach to enumerative geometry. Algebra i Analiz 17(2), 170–214 (2005) (English translation: St. Petersbg. Math. J. 17, 343–375 (2006))
Yang J.: Tropical Severi varieties. Port. Math. 70(1), 59–91 (2013)
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The author thanks Alexander Esterov and Eric Katz for useful discussions. Also the author is very thankful for the anonymous referee.
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Yang, J.J. Secondary fans and tropical Severi varieties. manuscripta math. 149, 93–106 (2016). https://doi.org/10.1007/s00229-015-0773-3
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DOI: https://doi.org/10.1007/s00229-015-0773-3