Gluing Affine Torus Actions Via Divisorial Fans

Abstract

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.

Euler sequence for complete smooth K⁎-surfaces

“…These varieties admit a polyhedral description given by K. Altmann and J. Hausen for the affine case in [AH06] and later, together with H. Suß, in [AHS08] for the non-affine case. In the following section, we briefly recall this construction as well as a definition by N. Ilten and H. Suß in [IS11] that helps to simplify the notation.…”

Families of invariant divisors on rational complexity-one T -varieties

“…We now recall the construction of T -varieties from p-divisors and divisorial fans, see [AH06] and [AHS08], as well as recalling the description of invariant Cartier divisors on complexity-one T -varieties [PS11]. For an introduction to and a survey of the theory of T -varieties, see [AIP + 12].…”

On intersection cohomology with torus actions of complexity one

“…Note that contraction-free T-varieties of complexity one were studied by Mumford in [KKMS73,Chapter IV]. These combinatorial descriptions admit a generalization to the setting of T-varieties (see [AH06,AHS08,Tim08,Lan14]). The description in [AH06] of an affine T-variety is in term of a divisor on a normal variety where its coefficients are polyhedra in N Q .…”

“…Such a combinatorial object is called a polyhedral divisor. More generally, the combinatorial description introduced in [AHS08] for a T-variety involves a divisorial fan which corresponds to a finite set of polyhedral divisors (with some additional conditions).…”

“…Preliminaries on T-varieties. In this section, we recall some basic notions on algebraic torus actions of complexity one (see [AH06,AHS08,Tim08,Lan14] for details).…”