Explicit Sarkisov program for regular surfaces over arbitrary fields and applications

Abstract

We prove the Sarkisov program for projective surfaces over excellent base rings, including the case of non-perfect base fields $k$ of characteristic $p>0$. We classify the Sarkisov links between Mori fibre spaces and their relations for regular surfaces, generalising work of Iskovskikh. As an application, we discuss rationality problems for regular surfaces and the structure of the plane Cremona group.

v3: Added a new section on Geiser and Bertini involutions, and fully characterised points in Sarkisov general position in the plane. Corrected minor inaccuracies