Appears in collection : Summer School 2025 - Cremona Group

Weil's regularization theorem asserts that for any rational action of an algebraic group G on a variety X\, there exists a variety on which G acts regularly and which is equivariantly birational to X. This theorem and its refinements are key ingredients for classifying algebraic subgroups of groups of birational transformations\, in classical work of Enriques and recent work of Blanc\, Zimmermann\, Fong and others. The lectures will first discuss the notions occurring in the regularization theorem\, and some of its applications. We will then present a proof of this theorem and further developments.