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

Abstract

Starting from ${\mathbb{C}}^{\ast }$-actions on complex projective varieties, we construct and investigate birational maps among the corresponding extremal 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 find explicit examples of rational homogeneous varieties admitting a ${\mathbb{C}}^{\ast }$-action whose weighted blow-up at the extremal fixed point components gives a birational map among two projective varieties which is locally a toric non-equalized flip.