First steps in tropical intersection theory

By Lars Allermann, Johannes Rau. 2007

191 citations recorded by the source corpus.

We establish first parts of a tropical intersection theory. Namely, we define cycles, Cartier divisors and intersection products between these two (without passing to rational equivalence) and discuss push-forward and pull-back. We do this first for fans in $${\mathbb{R}^{n}}$$ and then for “abstract” cycles that are fans locally. With regard to applications in enumerative geometry, we finally have a look at rational equivalence and intersection products of cycles and cycle classes in $${\mathbb{R}^{n}}$$ .

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

View Paper: https://www.semanticscholar.org/paper/1a9e18b70c32fe253a7a743195bb367cfdc456d0

How researchers see this paper

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

In Algebra and Number Theory

In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

Descendant log Gromov-Witten invariants for toric varieties and tropical curves (2016, background): “ribe here how to generalize to allow for arbitrary algebraic cycles. In Deﬁnition 2.9, in place of codimension ai aﬃne subspace Ai ⊆ NQ, allow Ai to be any tropical polyhedral complex in the sense of [AR10]. In place of codimension bj aﬃne subspaces Bj ⊆ NQ, we DESCENDANT LOG GROMOV-WITTEN INVARIANT”
Matroid Theory for Algebraic Geometers (2014, unknown): “(∆) and ι∗α = c1(L), the ﬁrst Chern class of L. If c ∈ Ak(X(∆)) is viewed as a Minkowski weight, we may compute the cup product ι∗α∪c as an element of Ak+1(X) by using a formula that ﬁrst appeared in [5]: for σ ∈ ∆(k),τ ∈ ∆(k+1), let u σ/τ be a vector in Nσ descending to vσ/τ in N/Nτ; then the value”
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 [8] and studied in the context of tropical intersection theory by Mikhalkin [16], Allermann–Rau [3], and the second author [10,11]”
MODULI SPACES OF RATIONAL WEIGHTED STABLE CURVES AND TROPICAL GEOMETRY (2014, unknown): “, φr on Y and C := φ1 · · · · · φr · Y (for an in-depth discussion of rational functions and divisors, see for example [2]).”
Tropical descendant Gromov–Witten invariants (2008, methodology): “morphism, which is not required in [ 1 ], makes sure that we do not have to refine X further.”
Tropical intersection theory from toric varieties (2009, background): “eneral tropical intersection theory of tropical polyhedral complexes in Rn. There are two deﬁnitions of tropical intersection product: one given by Mikhalkin in [14] and one given by Allermann-Rau in [1]. In this section, we prove that they are equal. A polyhedral complex C in NR is a ﬁnite collecti”
A guide to tropicalizations (2011, methodology): “with rational equivalence of cycles, this plays no role here. 13.16 We will explain how the linear map f:= Trop v (’) : N R !N0 induces a push{forward map of tropical cycles. For details, we refer to [AR], x7. Let C be a tropical cycle in N R of pure dimension d. After a subdivision of C, we may ass”