On certain K-equivalent birational maps (Q2633094)

Flops are basic and important objects in birational geometry, typical examples are standard flops and twisted Mukai flops. Motivated by those examples, the author investigates simple birational maps. A simple birational map (probably not a good name) is a K-equivalent birational map between smooth projective varieties which can be resolved by one blowing up along a smooth center of each side. As the main result, this paper gives a criterion for a simple birational map to be a stadard flop or a twisted Mukai flop. To be more precise, let $\theta:X\dashrightarrow X^+$ be a simple birational map and $P, P^+$ be the centers. Let $E$ be the exceptional divisor of the blowing ups (which can be shown to be the same). Assume further that $P$ and $P^+$ admit projective bundle structures over a common variety $C$, and the induced maps $E\to P\to C$ and $E\to P^+\to C$ coincides, then $\theta$ is a standard flop or a twisted Mukai flop. As an application, this paper gives a classification of simple birational maps of dimension up to $5$, any such simple birational map is a standard flop, or a twisted Mukai flop, or an Abuaf flop.