Equivariant minimal model program

Abstract

The purpose of the survey is to systematize a vast amount of information about the minimal model program for varieties with group actions. We discuss the basic methods of the theory and give sketches of the proofs of some principal results. Bibliography: 243 titles.

Embeddings of the symmetric groups to the space Cremona group

“…Throughout this paper we use the terminology and notation of the equivariant minimal model program [Pro21]. In particular, a GQ-Fano variety X is a variety equipped with an action of a finite group G such that X has only terminal GQ-factorial singularities, rk PicpXq " 1, and the anticanonical divisor ´KX is ample.…”

“…Assume that dimpZq ě 1. Then f is either GQ-del Pezzo fibration and Z » P 1 or a GQ-conic bundle and Z a rational surface (see [Pro21,§ 10]). The map f induces a homomorphism Φ : G Ñ AutpZq.…”

“…Recall that a normal G-variety X is said to be GQ-factorial if some multiple nD of any G-invariant divisor is Cartier [Pro21].…”

Finite groups of birational transformations

“…This is the basic tool in the study of finite groups of birational transformations. For a detailed exposition we refer to [78].…”

“…The following assertions are usually formulated for varieties without group actions. The corresponding equivariant versions can be easily deduced from non-equivariant ones (see [78]). The conjecture is known to be true in dimension ≤ 4 [68,92], as well as in the case where K X is big [13], and in some other cases.…”