Tropical intersection theory from toric varieties

By Eric Katz. 2009

75 citations recorded by the source corpus.

We apply ideas from intersection theory on toric varieties to tropical intersection theory. We introduce mixed Minkowski weights on toric varieties which interpolate between equivariant and ordinary Chow cohomology classes on compact toric varieties. These objects fit into the framework of tropical intersection theory developed by Allermann and Rau. Standard facts about intersection theory on toric varieties are applied to show that the definitions of tropical intersection product on tropical cycles in $${\mathbb{R}^n}$$ given by Allermann–Rau and Mikhalkin are equivalent. We introduce an induced tropical intersection theory on subvarieties on a toric variety. This gives a conceptual proof that the intersection of tropical ψ-classes on $${\overline{\mathcal{M}}_{0,n}}$$ used by Kerber and Markwig computes classical intersection numbers.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f489c52c28e13cdc9719ce53e982c57d63a4d1e8

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

Matroid Theory for Algebraic Geometers (2014, methodology): “The following lemma can be proved using intersection theory [51] or by elementary means [5, Prop 3.”
Log-concavity of characteristic polynomials and the Bergman fan of matroids (2011, background): “n 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]. We use the fact that there is an explicit Poincar´edual to a compactiﬁcation of the comple”