Generalized Minkowski weights and Chow rings of T-varieties

TL;DR

This paper provides a combinatorial framework for understanding the Chow cohomology and intersection theory of certain T-varieties, enabling new computational methods for intersection numbers.

Contribution

It introduces generalized Minkowski weights to characterize Chow cohomology of rational T-varieties of complexity one, expanding combinatorial tools in algebraic geometry.

Findings

Characterization of Chow cohomology groups via Minkowski weights

Description of intersection products with Cartier divisors

New combinatorial method for computing intersection numbers

Abstract

We give a combinatorial characterization of Fulton's operational Chow cohomology groups of a complete , $\Q$-factorial, rational T-variety of complexity one in terms of so called generalized Minkowsky weights in the contraction-free case. We also describe the intersection product with Cartier invariant divisors in terms of the combinatorial data. In particular this provides a new way of computing top intersection numbers of invariant Cartier divisors combinatorially.