On restriction of roots on affine T-varieties

TL;DR

This paper studies how roots of a normal affine T-variety relate to roots of the larger torus action, providing new insights and elementary proofs for the structure of roots in affine Cremona groups and toric varieties.

Contribution

It proves that roots with respect to a subtorus can be derived from roots with respect to the larger torus, offering a new approach to understanding root structures in affine T-varieties.

Findings

Roots of X with respect to T are restrictions of roots with respect to .

Elementary proof for the description of roots of the affine Cremona group.

Results on root restrictions in affine toric varieties.

Abstract

Let X be a normal affine algebraic variety with regular action of a torus \TT and T\subset\TT be a subtorus. We prove that each root of X with respect to T can be obtained by restriction of some root of X with respect to \TT. This allows to get an elementary proof of the description of roots of the affine Cremona group. Several results on restriction of roots in the case of subtorus action on an affine toric variety are obtained.