Cremona Group

The group of birational self-transformations of projective n-space over a fixed field, equivalently the field automorphisms of a purely transcendental function field.

Version

v1 · 2026-08-30 · History

Domain-specific #

1588

Origin domain

birational geometry

Subdomain

groups of birational transformations

Aliases

Cremona transformation group

Core Idea

Fix a field \(k\) and a positive integer \(n\). The Cremona group of rank \(n\) over \(k\), customarily written \(\operatorname{Cr}_n(k)\), is the group of birational self-transformations of projective \(n\)-space:

\[ \operatorname{Cr}_n(k)=\operatorname{Bir}_k(\mathbb P^n_k). \]

Equivalently, after choosing affine coordinates, it is the group of \(k\)-automorphisms of the purely transcendental function field:

\[ \operatorname{Cr}_n(k)\cong \operatorname{Aut}_k k(x_1,\ldots,x_n). \]

These are two presentations of the same object. A birational map is defined on a dense open part of projective space and has a rational inverse on another dense open part; it need not be defined, regular, or one-to-one at every point.

Scope of Application

The primary scope is birational geometry. A rational variety \(X\) of dimension \(n\) is birational to \(\mathbb P^n_k\); a chosen birational map transports \(\operatorname{Bir}_k(X)\) into a conjugate presentation of \(\operatorname{Cr}_n(k)\). This lets geometers replace coordinate-heavy rational transformations by one ambient group while keeping track of how different models resolve indeterminacy.

In plane geometry, the group organizes classical factorization. Over an algebraically closed field, the Noether–Castelnuovo theorem generates \(\operatorname{Cr}_2(k)\) from \(\operatorname{PGL}_3(k)\) and the standard quadratic involution.

Clarity

Use five checks. First, state \(k\) and \(n\). Second, identify either \(\mathbb P^n_k\) or \(k(x_1,\ldots,x_n)\). Third, verify rational invertibility, not merely dominance. Fourth, distinguish equality on dense opens from equality of one chosen coordinate tuple. Fifth, name whether a claim concerns abstract group structure, a topology on the group, degree-filtered parameter spaces, or an action on a birational model.

Manages Complexity

The construct compresses innumerable rational coordinate substitutions into one compositional system. Instead of treating each formula independently, one can ask whether it lies in a standard subgroup, how it factors into generators, which conjugacy class contains it, what its iterates do, and which invariants survive a change of birational coordinates.

Abstract Reasoning

1. If a homogeneous presentation has degree one, its element lies in \(\operatorname{PGL}_{n+1}(k)\); the converse holds for regular automorphisms of projective space. 2. If a rational self-map lacks a rational inverse, closure in a transformation semigroup may hold, but membership in the Cremona group fails. 3. If \(f\) and \(g\) are represented by coordinate tuples, the raw degree of \(f\circ g\) can be below \(\deg(f)\deg(g)\) because substitution may create a common factor that must be cancelled.

Knowledge Transfer

Exact transfer occurs among rational models of the same function field. If \(X\) is \(k\)-rational and \(\phi:X\dashrightarrow\mathbb P^n_k\) is birational, then conjugation by \(\phi\) identifies \(\operatorname{Bir}_k(X)\) with a presentation of \(\operatorname{Cr}_n(k)\). The isomorphism depends on the chosen birational identification, while conjugacy-invariant conclusions do not.

Many tools transfer to other birational automorphism groups: resolve indeterminacy, study induced actions on divisor groups, classify elements by growth, and analyze subgroups through preserved fibrations or geometric models.

Relationships to Other Abstractions

Current abstraction Cremona Group Domain-specific

Parents (1) — more general patterns this builds on

• Cremona Group is a kind of Group Prime

  The minimal currently valid prospective parent is the live prime Group.

Hierarchy paths (5) — routes to 5 parentless roots

• Cremona Group → Group → Monoid → Semigroup → Set and Membership

Show alternative paths (4)

Neighborhood in Abstraction Space

Cremona Group sits in a sparse region of the domain-specific corpus (85^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Fields, Norms & Birational Groups (7 abstractions)

Nearest neighbors

• Norm Form — 0.80
• Algebraic Cycle — 0.80
• Quadratic Field — 0.80
• Profinite Integer — 0.80
• Euler sequence — 0.80

Computed from structural-signature embeddings · 2026-09-08