On certain K-equivalent birational maps

Abstract

We study K-equivalent birational maps which are resolved by a single blowup. Examples of such maps include standard flops and twisted Mukai flops. We give a criterion for such maps to be a standard flop or a twisted Mukai flop. As an application, we classify all such birational maps up to dimension 5.

On the Abuaf-Ueda Flop via Non-Commutative Crepant Resolutions

“…Recently, Kanemitsu [10] classified simple flops of dimension up to eight, which is a certain generalization of a theorem of Li [15]. His list contains many examples of flops for which the Bondal-Orlov conjecture is still open.…”

Mukai pairs and simple $K$-equivalence

“…In this paper, we will focus on a class of K-equivalent birational maps, called simple K-equivalent maps. A K-equivalent map is called simple, if we can choose a resolution as above such that f i are smooth blow-ups [Li18]. At a first glance, the assumption in this definition seems to be too strong.…”

“…Based on the above interesting phenomena, it is natural to wonder further examples of simple K-equivalent birational maps, and try to classify these birational maps. Such an attempt is started by [Li18], and it is proved that simple Kequivalent maps in dimension at most 5 are only three types; standard flops, Mukai flops and Abuaf's flop. Also it is desirable to have a nice structure theorem for simple K-equivalent maps.…”

“…Acknowledgements. The author wishes to express his gratitude to Professor Yoichi Miyaoka for suggesting me the relation between Mukai pairs and flops, and to Professor Shigeru Mukai for drawing the authors's attention to [Li18] and for his insightful comments and discussions. The author is also grateful to Professor Yuki Hirano for his helpful comments and telling me about the paper [Ued18], and to Doctor Sho Ejiri for helpful discussions on a variant of Theorem 2.2.…”

On the Abuaf-Ueda Flop via Non-Commutative Crepant Resolutions

“…Recently, Kanemitsu [Kan18] classified simple flops of dimension up to eight, which is a certain generalization of the theorem of Li [Li17]. It is interesting to prove the derived equivalence for all simple flops that appear in Kanemitsu's list using tilting bundles, and we can regard this article as a part of such a project.…”