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-Włodarczyk 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.

中文翻译：

---

Morelli-Włodarczyk 共边主义和屋顶翻转的例子

我们引入了屋顶翻转的概念，即正常射影簇的一个小修改，它是由允许两个射影丛结构的皮卡德2号平滑射影簇建模的。例子包括 Atiyah 翻牌和 Mukai 翻牌，分别由\(\mathbb {P}^1\times \mathbb {P}^1\)和\(\mathbb {P}\left( T_{\mathbb { P}^2}\右）\）。使用 Morelli-Włodarczyk 协边主义，我们证明了皮卡德数 1 的任何平滑射影变体，赋予仅具有两个定点分量的\({\mathbb C}^*\)作用，都会引起屋顶翻转。