A Tropical Toolkit

By Eric Katz. 2006

87 citations recorded by the source corpus.

Abstract We give an introduction to Tropical Geometry and prove some results in tropical intersection theory. The first part of this paper is an introduction to tropical geometry aimed at researchers in Algebraic Geometry from the point of view of degenerations of varieties using projective not-necessarily-normal toric varieties. The second part is a foundational account of tropical intersection theory with proofs of some new theorems relating it to classical intersection theory.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ba48ceb0375e6cd4f35472e310e31eb5df80d5b4

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Citations in the corpus

Matroid Theory for Algebraic Geometers (2014, background): “We recommend [38] as a reference for toric varieties, [39] for results on intersection theory on toric varieties, and [50] for results most directly suited to our purposes.”
Fibers of tropicalization (2007, methodology): “If RK was a discrete valuation ring then this gap could be filled by applying standard arguments from intersection theory to Spec RK [x]/I , as in [7].”
A guide to tropicalizations (2011, methodology): “a certain complete -rational complex which has a subcomplex with support equal to Trop v (X). At the end, we relate the Grobner complex to the initial degenerations of X. This section is inspired by [Kat], Section 5, which in turn was in uenced by Tevelev. We work here with more general base elds, b”
Log-concavity of characteristic polynomials and the Bergman fan of matroids (2011, methodology): “To identify the coefficients as intersection numbers, we use the combinatorial interpretation of the intersection theory on toric varieties developed by Fulton-Sturmfels [9] and studied in the context of tropical intersection theory by Mikhalkin [17], AllermannRau [3], and the second author [11, 12]”
On the tropical Torelli map (2009, background): “For an introduction to tropical geometry, see the surveys [39,55,52,22,41,28,43], or the books in preparation [44,34].”