Torus Invariant Curves

TL;DR

This paper studies torus invariant curves on complete normal varieties with codimension-one torus actions, providing combinatorial formulas for intersections and generalizing the toric cone theorem to complexity-one T-varieties.

Contribution

It introduces combinatorial formulas for intersections of invariant divisors and curves and extends the toric cone theorem to complexity-one T-varieties.

Findings

Derived combinatorial formulas for intersection calculations.

Generalized the toric cone theorem to complexity-one T-varieties.

Provided a new framework for understanding invariant curves in T-varieties.

Abstract

Using the language of T-varieties, we study torus invariant curves on a complete normal variety $X$ with an effective codimension-one torus action. In the same way that the $T$-invariant Weil divisors on $X$ are sums of "vertical" divisors and "horizontal" divisors, so too is each $T$-invariant curve a sum of "vertical" curves and "horizontal" curves. We give combinatorial formulas that calculate the intersection between $T$-invariant divisors and $T$-invariant curves, and generalize the celebrated toric cone theorem to the case of complete complexity-one $T$-varieties.