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On torus actions of higher complexity. (English) Zbl 1445.14067

For complete algebraic varieties \(X\) with finitely generated \(\operatorname{Pic}(X)\) one can define the so-called Cox ring \(R(X)\). It is multigraded via \(\operatorname{Pic}(X)\), and if \(R(X)\) is finitely generated (\(X\) is a “Mori-Dream space” = “MDS”), then \(X\) can be recovered from \(R(X)\) as a torus quotient. To obtain this, a certain irrelevant subspace of \(\operatorname{Spec} R(X)\) has to be removed, and it is the relationship between the combinatorics of these subspaces (described by “bunches of cones” or their Gale duals) and the birational geometry of \(X\) which makes this approach very fruitful.
Understanding \(X\) as a quotient of \(\operatorname{Spec} R(X)\) makes it possible to embed \(\operatorname{Spec} R(X)\) equivariantly, i.e.\(\operatorname{Pic}(X)\)-homogeneously, into some affine space and to obtain, via taking quotients, a special embedding of \(X\) into a toric variety. The authors call those embedded varieties “explicit”.
In the present paper, they generalize this construction to varieties \(X\) being already equipped with some other torus action of a certain complexity.
If \(X\) is MDS, it is possible to proceed as before, but under preserving this additional action. This generalizes earlier work for complexity one.
The theory is applied to the special case of so-called general arrangement varieties, in particular those of Picard number at most two. Moreover, the methods allow some contribution to the classification of (almost) Fano varieties with torus action.

MSC:

14L30 Group actions on varieties or schemes (quotients)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14J45 Fano varieties

Software:

MDSpackage
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

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