Tropical intersection theory from toric varieties
The Chow cohomology group \(A^{k}(X(\Delta))\), of a toric variety \(X(\Delta)\), is a group of certain functions (called \textit{Minkowski weights}) on the set of codimension \(k\) cones in the fan \(\Delta\). A closed sub-variety \(V\) of a toric variety \(X(\Delta)\), satisfying some additional conditions, gives rise to a weighted polyhedral sub-complex \(\mathrm{Trop}(V)\) of \(\Delta\). This polyhedral complex is a tropical cycle in the sense of tropical geometry. Given a sub-variety \(V\) satisfying requisite conditions, let \(c_{V}\) denote its Minkowski weight (which is Poincaré-dual to the fundamental class \([V]\)) and \(\mathrm{Trop}(V)\) the tropical cycle. In this paper the author studies the relation between \(c_{V}\) and \(\mathrm{Trop}(V)\) and shows that the tropical intersection product outlined by \textit{G. Mikhalkin} [in: Proceedings of the 13th Gökova geometry-topology conference, Gökova, Turkey, May 28--June 2, 2006. Cambridge, MA: International Press. 39--51 (2007; Zbl 1203.14027)] equals the cup product in Chow cohomology. \textit{L. Allermann} and \textit{J. Rau} [Math. Z. 264, No. 3, 633--670 (2010; Zbl 1193.14074)] outlined another approach to tropical intersection theory by introducing tropical Cartier divisors, associated tropical Weil divisors, rational equivalence. The author shows that these constructions can also be described in Chow cohomology using \textit{mixed Minkowski weight}. As a result, the equivalence of these two tropical intersection theories is easily deduced from the standard properties (pushforward) of Chow cohomology. The equivalence of these intersection theories does not follow from direct tropical computations. This point of view is used to give a new calculation of the intersection numbers of \(\psi\) classes in the moduli space of genus zero rational curves using tropical geometry. This was previously computed by \textit{M. Kerber} and \textit{H. Markwig} [Int. Math. Res. Not. 2009, No. 2, 221--240 (2009; Zbl 1205.14070)].
scientific article