First steps in tropical intersection theory

Abstract

ABSTRACT. 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 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 R n .

Moduli Spaces of Rational Weighted Stable Curves and Tropical Geometry

“…We now investigate the tropical analogue. [2]). Then we define the pullback of C along Y to be by (1, .…”

Tropical descendant Gromov–Witten invariants

“…First of all, let us briefly recall the constructions from [1] that we need here: A cycle X is a balanced (weighted, pure-dimensional, rational and polyhedral) complex (resp. fan) in R n .…”

“…Here, a primitive vector v σ/τ of σ modulo τ is a integer vector in Z n that points from τ towards σ and fulfills the primitive condition: The lattice Zv σ/τ +(V τ ∩Z n ) must be equal to the lattice V σ ∩Z n . Slightly differently, in [1] the class of v σ/τ modulo V τ is called primitive vector and v σ/τ is just a representative of it. For us, a polyhedron σ is always understood to be closed.…”

“…It states that ev 1 * (F) has only standard outer directions −e 1 , −e 2 and e 1 + e 2 . Knowing this, we push forward ev * 1 (G 1 ) · F via ev 1 , which does not change the degree and use the projection formula [1]. It tells us that ev 1 * (ev…”

Tropical Hurwitz Numbers

“…In particular, they are known to be tropical varieties. That allows us to use tropical intersection products on these spaces (see [Mik06] or [AR07]). We can intersect Psi-classes as in [KM07].…”