We investigate the Cox ring of a normal complete variety X with algebraic torus action. Our first results relate the Cox ring of X to that of a maximal geometric quotient of X. As a consequence, we obtain a complete description of the Cox ring in terms of generators and relations for varieties with torus action of complexity one. Moreover, we provide a combinatorial approach to the Cox ring using the language of polyhedral divisors. Applied to smooth K * -surfaces, our results give a description of the Cox ring in terms of Orlik-Wagreich graphs. As examples, we explicitly compute the Cox rings of all Gorenstein del Pezzo K * -surfaces with Picard number at most two and the Cox rings of projectivizations of rank two vector bundles as well as cotangent bundles over toric varieties in terms of Klyachko's description.
Generalizing the passage from a fan to a toric variety, we provide a combinatorial approach to construct arbitrary effective torus actions on normal, algebraic varieties. Based on the notion of a "proper polyhedral divisor" introduced in earlier work, we develop the concept of a "divisorial fan" and show that these objects encode the equivariant gluing of affine varieties with torus action. We characterize separateness and completeness of the resulting varieties in terms of divisorial fans, and we study examples like C * -surfaces and projectivizations of (nonsplit) vector bundles over toric varieties.
We consider Fano manifolds admitting an algebraic torus action with general orbit of codimension one. Using a recent result of Datar and Székelyhidi, we effectively determine the existence of Kähler-Ricci solitons for those manifolds via the notion of equivariant K-stability. This allows us to give new examples of Kähler-Einstein Fano threefolds, and Fano threefolds admitting a non-trivial Kähler-Ricci soliton.
We propose a method to compute a desingularization of a normal affine variety X endowed with a torus action in terms of a combinatorial description of such a variety due to Altmann and Hausen. This desingularization allows us to study the structure of the singularities of X. In particular, we give criteria for X to have only rational, (Q-)factorial, or (Q-)Gorenstein singularities. We also give partial criteria for X to be Cohen-Macaulay or log-terminal.Finally, we provide a method to construct factorial affine varieties with a torus action. This leads to a full classification of such varieties in the case where the action is of complexity one.
Using the language of polyhedral divisors and divisorial fans we describe
invariant divisors on normal varieties X which admit an effective codimension
one torus action. In this picture X is given by a divisorial fan on a smooth
projective curve Y. Cartier divisors on X can be described by piecewise affine
functions h on the divisorial fan S whereas Weil divisors correspond to certain
zero and one dimensional faces of it. Furthermore we provide descriptions of
the divisor class group and the canonical divisor. Global sections of line
bundles O(D_h) will be determined by a subset of a weight polytope associated
to h, and global sections of specific line bundles on the underlying curve Y.Comment: 16 pages; 5 pictures; small changes in the layout, further typos
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