Segre classes on smooth projective toric varieties
type: Item
instance of: scholarly article
title: Segre classes on smooth projective toric varieties
zbMATH Open document ID: 1310.14015
author: Torgunn Karoline Moe
author: Nikolay Qviller
published in: Mathematische Zeitschrift
publication date: 2013-11-11T00:00:00Z
full work available at URL: https://arxiv.org/abs/1204.4884
full work available at URL: http://hdl.handle.net/10852/37197
review text: 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.
Mathematics Subject Classification ID: 14C17
Mathematics Subject Classification ID: 14M25
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14Q99
zbMATH DE Number: 6224346
zbMATH Keywords: Segre classes
zbMATH Keywords: toric varieties
zbMATH Keywords: computational algorithm
zbMATH Keywords: nef cone
zbMATH Keywords: intersection theory
reviewed by: Mateusz Michalek
describes a project that uses: CSM-A
describes a project that uses: Macaulay2
describes a project that uses: SageMath
MaRDI profile type: Publication
OpenAlex ID: W3102898181
arXiv ID: 1204.4884
cites work: Computing characteristic classes of projective schemes.
cites work: Q4324883
cites work: Toric varieties
cites work: THE GEOMETRY OF TORIC VARIETIES
cites work: A METHOD TO COMPUTE SEGRE CLASSES OF SUBSCHEMES OF PROJECTIVE SPACE
cites work: Q3328651
cites work: Introduction to Toric Varieties. (AM-131)
cites work: Q4143433
cites work: Multigraded Castelnuovo-Mumford regularity
cites work: Liaison des variétés algébriques. I
DOI: 10.1007/S00209-013-1146-9
description: scientific article; zbMATH DE number 6224346