Morelli-Włodarczyk cobordism and examples of rooftop flips

Abstract

We introduce the notion of rooftop flip, namely a small modification among normal projective varieties which is modeled by a smooth projective variety of Picard number 2 admitting two projective bundle structures. Examples include the Atiyah flop and the Mukai flop, modeled respectively by ${\mathbb{P}}^1\times {\mathbb{P}}^1$ and by $\mathbb{P}\left(T_{{\mathbb{P}}^2}\right)$. Using the Morelli-Wlodarczyk cobordism, we prove that any smooth projective variety of Picard number 1, endowed with a ${\mathbb{C}}^{\ast }$-action with only two fixed point components, induces a rooftop flip.

15 pages, 2 figures. Comments are very welcome!