Toric non-equalized flips associated to $\mathbb{C}^*$-actions

TL;DR

This paper explores birational maps arising from $ ext{C}^*$-actions on complex projective varieties, focusing on toric non-equalized flips and their relation to the action's properties, providing explicit examples involving rational homogeneous varieties.

Contribution

It introduces and studies the concept of toric non-equalized flips associated with $ ext{C}^*$-actions, linking them to non-equalized actions and providing explicit examples involving rational homogeneous varieties.

Findings

Toric non-equalized flips are characterized and related to $ ext{C}^*$-actions.

Explicit examples of rational homogeneous varieties with such flips are constructed.

Weighted blow-ups at extremal fixed points produce birational maps that are toric non-equalized flips.

Abstract

Starting from $\mathbb{C}^*$-actions on complex projective varieties, we construct and investigate birational maps among the corresponding extremal fixed point components. We study the case in which such birational maps are locally described by toric flips, either of Atiyah type or so called non-equalized. We relate this notion of toric flip with the property of the action being non-equalized. Moreover, we find explicit examples of rational homogeneous varieties admitting a $\mathbb{C}^*$-action whose weighted blow-up at the extremal fixed point components gives a birational map among two projective varieties that is locally a toric non-equalized flip.