On the Genus of Birational Maps Between Threefolds

Abstract

In this note we present two equivalent definitions for the genus of a birational map $\varphi: X --\rightarrow Y$ between smooth complex projective threefolds. The first one is the definition introduced by Frumkin [Mat. Sb. (N.S.) 90(132):196–213, 325, 1973], and the second one was recently suggested to me by S. Cantat. By focusing first on proving that these two definitions are equivalent, one can obtain all the results in M.A. Frumkin [Mat. Sb. (N.S.) 90(132):196–213, 325, 1973] in a much shorter way. In particular, the genus of an automorphism of $\mathbb{C}^{3}$, view as a birational self-map of $\mathbb{P}^{3}$, will easily be proved to be 0.