Segre classes on smooth projective toric varieties
For a given scheme \(X\) the Segre class \(s(Z,X)\) is an important invariant of a subscheme \(Z\). It belongs to the Chow group of \(Z\), or by the natural push-forward to the Chow group of \(X\). The Segre class plays a very important role in intersection theory -- cf. for example [\textit{W. Fulton}, Intersection theory. 2nd ed. Berlin: Springer (1998; Zbl 0885.14002), Chapter 4]. The authors present an algorithm to compute the Segre class of any closed subscheme of a smooth projective toric variety. This generalizes (and was inspired by) an algorithm presented for projective spaces by Eklund, Jost and Peterson [\textit{D. Eklund} et al., J. Algebra Appl. 12, No. 2, Paper No. 1250142, 15 p. (2013; Zbl 1274.13044)]. The input of the algorithm is the fan of the variety \(X\) and the generators of the ideal in the Cox ring of \(X\) that defines \(Z\) (recall that there is a 1-1 correspondence between saturated, homogeneous ideals in the Cox ring and closed subschemes of \(X\)). The algorithm heavily relies on computation of saturation (that can often be difficult) and choice of random elements at some points. The authors implement the algorithm using Macaulay2 and Sage. The examples of Hirzebruch surfaces and the product \(\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1\) are worked out in details.
scientific article; zbMATH DE number 6224346