Fano threefolds with 2-torus action
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.
scientific article; zbMATH DE number 6359324