On the genus of birational maps between 3-folds

TL;DR

This paper compares two definitions of the genus of a birational map between smooth complex projective 3-folds, proving their equivalence and simplifying existing results, including automorphisms of a73.

Contribution

It establishes the equivalence of two definitions of genus for birational maps between 3-folds, streamlining proofs and clarifying genus for automorphisms of a73.

Findings

The two definitions of genus are equivalent.

Automorphisms of a73 have genus 0.

Simplified proof of genus properties for birational maps.

Abstract

In this note we present two equivalent definitions for the genus of a birational map X --> Y between smooth complex projective 3-folds. The first one is the definition introduced in 1973 by M. A. Frumkin, 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 of the paper of Frumkin in a much shorter way. In particular, the genus of an automorphism of $\mathbb{C}^3$, view as a birational self-map of the projective space, will easily be proved to be 0.