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

v2 · 2026-09-19 · History

Domain-specific #

1588

Domain group

Formal Sciences

Origin domain

Mathematics

Subdomains

Birational Geometry, Groups of Birational Transformations → Mathematics

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.

How would you explain it like I'm…

Move-and-Undo Club

Imagine rules made from math formulas that slide and stretch all the dots in a space to new spots. Each rule has an undo rule that puts them back, though at a few special spots the rule gets stuck and can't say where a dot goes. The Cremona group is the big collection of all these move-and-undo rules.

Reversible Formula Moves

Mathematicians like to study ways of moving the points of a space around using formulas made of fractions, where another such formula can undo the move. A formula like that might break at a few special spots, a bit like dividing by zero, and that's allowed, as long as it works almost everywhere. The Cremona group is the collection of all such undoable formula-moves on one kind of space called projective space. Doing one move and then another gives another move in the collection, and every move has an undo move, which is what makes it a "group."

Birational Maps of Projective Space

In algebra, a group is a collection of transformations you can combine and undo. The Cremona group collects all "birational" transformations of projective space: maps given by fractions of polynomials that have an inverse of the same kind. Unlike ordinary rearrangements, these maps don't have to be defined at every point or be one-to-one everywhere; they only need to work and be reversible on a large open part of the space. Two formulas that agree on such a large part count as the same map. Combining maps means substituting one formula into another and cancelling common factors. Which number system you use (the field) and how many dimensions you have are part of the definition; changing either one changes the group.

 

Fix a field k and a positive integer n. The Cremona group Cr_n(k) is the group of birational self-maps of projective n-space over k, equivalently the group of k-automorphisms of the rational function field k(x_1, ..., x_n); these are two presentations of the same object. A birational map is defined on a dense open subset and has a rational inverse defined on another dense open subset; it need not be regular or injective everywhere. In homogeneous coordinates, an element is given by n+1 homogeneous polynomials of equal degree with no common nonconstant factor, such that the induced rational map has a rational inverse, and representatives agreeing on a dense open set define the same element. The group law is composition: substitute formulas and cancel common factors, with the identity map as neutral element and the birational inverse as group inverse. Thus Cr_n(k) gathers all rational coordinate changes of one projective space into a single algebraic object. Researchers use it to factor birational maps, classify finite and algebraic subgroups, compare conjugacy classes, study degree growth, regularize group actions on birational models, and test algebraic or topological simplicity. Changing k or n can change the elements, subgroups, conjugacy classes, generation results, and topology.

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) — outgoing typed relations

• Cremona Group is a kind of Group Prime

  The minimal currently valid the broader abstraction is the live prime Group.

Hierarchy paths (5) — routes to 5 parentless roots

• Cremona Group → is a kind of Group → is a kind of Monoid → is a kind of Semigroup → is a kind of 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 — Unclustered & Miscellaneous (2769 abstractions)

Nearest neighbors

• Algebraic Variety — 0.84
• Grothendieck–Riemann–Roch theorem — 0.81
• N-Square Identity — 0.81
• Norm Form — 0.80
• Algebraic Cycle — 0.80

Computed from structural-signature embeddings · 2026-10-25