Gluing Affine Torus Actions Via Divisorial Fans

By K. Altmann, J. Hausen, Hendrik Süß. 2006

125 citations recorded by the source corpus.

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 $ \mathbb{C} $*-surfaces and projectivizations of (nonsplit) vector bundles over toric varieties.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/2fbe4cc7160421857f377c7536a58d56d239b621

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. Lars Petersen
2. Chi Li
3. Hendrik Süß

In Discrete Mathematics and Combinatorics

1. Lars Petersen
2. Chi Li
3. Hendrik Süß

In Geometry and Topology

1. Chi Li
2. Lars Petersen
3. Hendrik Süß
4. H. Blum
5. Yuchen Liu
6. Ziquan Zhuang
7. Mattias Jonsson
8. Xiaowei Wang

Citations in the corpus

• Torus invariant divisors (2008, methodology): “Its divisorial fan S over P 1 looks like Figure 4; cf. [ AHS08 , 8.5].”
• NORMAL SINGULARITIES WITH TORUS ACTIONS (2010, unknown): “For details see [AHS08].”
• Geodesic rays and stability in the cscK problem (2020, background): “By the structure theory of T-varieties (see [1]), X is birationally a torus fibration over the Chow quotient of X by T which will be denoted by X//T.”