Juergen Hausen, Christoff Hische, Milena Wrobel
Published 2018-02-01Version 1
We extend the Cox ring based combinatorial theory for rational varieties with torus action of complexity one to Mori dream spaces with torus action of arbitrary high complexity. The key idea is to work over the maximal orbit quotient, which keeps finite generation of the Cox ring. As a sample class we investigate Mori dream spaces with a projective space as maximal orbit quotient having a general hyperplane arrangement as critical locus. Here we obtain simply structured resulting Cox rings which in turn allows explicit classifcations, for example of the smooth Fano examples of complexity two and Picard number two.
Gorenstein Fano threefolds of Picard number one with a $\mathbb{K}^*$-action and maximal orbit quotient $\mathbb{P}_2$
Smooth projective varieties with a torus action of complexity 1 and Picard number 2
Normal singularities with torus actions
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