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 $\mathbb{ C}^*$-action with only two fixed point components, induces a rooftop flip.
Categories: math.AG