Fano threefolds with 2-torus action
type: Item
instance of: scholarly article
title: Fano threefolds with 2-torus action
zbMATH Open document ID: 1341.14021
published in: Documenta Mathematica
publication date: 2014-10-21T00:00:00Z
full work available at URL: https://arxiv.org/abs/1308.2379
full work available at URL: 30.html
reviewed by: Klaus Altmann
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 14J45
Mathematics Subject Classification ID: 32Q20
Mathematics Subject Classification ID: 14C20
zbMATH DE Number: 6359324
zbMATH Keywords: torus action
zbMATH Keywords: polytopes
zbMATH Keywords: Fano varieties
zbMATH Keywords: Kähler-Einstein metrics
zbMATH Keywords: Cox rings
author: Hendrik Süss
MaRDI profile type: Publication
arXiv ID: 1308.2379
review text: The author scans Mori/Mukai's list of Fano threefolds for those admitting a two-dimensional torus action. For every hit, he gives an almost combinatorial description within the language of divisorial polytopes: Each variety is represented by a piecewise linear function from its moment polytope to the group of divisors on \(\mathbb P^1\) (being the torus quotient of the Fano threefold). Composed with the degree map, this yields the Duistermaat-Heckman measure.NEWLINENEWLINEThis explicit description makes it possible to calculate invariants like the Picard rank, the Fano degree, the Futaki character and to describe the Cox ring via explicit generators and equations. Moreover, in most cases, the author can conclude if a Kähler-Einstein metric exists or not.
description: scientific article; zbMATH DE number 6359324