Cremona group

In birational geometry, the Cremona group, named after Luigi Cremona, is the group of birational automorphisms of the ${\displaystyle n}$-dimensional projective space over a field ${\displaystyle k}$, also known as Cremona transformations. It is denoted by ${\displaystyle Cr(\mathbb {P} ^{n}(k))}$, ${\displaystyle Bir(\mathbb {P} ^{n}(k))}$ or ${\displaystyle Cr_{n}(k)}$.

Historical origins

The Cremona group was introduced by the italian mathematician Luigi Cremona (1863, 1865).^[1] However, some historians consider Isaac Newton as a "founder of the theory of Cremona transformations" through his work done two centuries before, in 1667 and 1687.^[2][3] Contributions were also made by Hilda Phoebe Hudson in the 1900s.^[4]

Basic properties

The Cremona group is naturally identified with the automorphism group ${\displaystyle \mathrm {Aut} _{k}(k(x_{1},...,x_{n}))}$ of the field of the rational functions in ${\displaystyle n}$ indeterminates over ${\displaystyle k}$. Here, the field ${\displaystyle k(x_{1},...,x_{n})}$ is a pure transcendental extension of ${\displaystyle k}$, with transcendence degree ${\displaystyle n}$.

The projective general linear group ${\displaystyle \mathrm {PGL} _{n+1}}$ is contained in ${\displaystyle Cr_{n}}$. The two are equal only when ${\displaystyle n=0}$ or ${\displaystyle n=1}$, in which case both the numerator and the denominator of a transformation must be linear.^[5]

A longlasting question from Federigo Enriques concerns the simplicity of the Cremona group. It has been now mostly answered.^[6]

The Cremona group in 2 dimensions

In two dimensions, Max Noether and Guido Castelnuovo showed that the complex Cremona group is generated by the standard quadratic transformation, along with ${\displaystyle \mathrm {PGL} (3,k)}$, though there was some controversy about whether their proofs were correct. Gizatullin (1983) gave a complete set of relations for these generators. The structure of this group is still not well understood, though there has been a lot of work on finding elements or subgroups of it.

• Cantat & Lamy (2010) showed that for an algebraicly closed field ${\displaystyle k}$, the group ${\displaystyle Cr_{2}(k)}$ is not simple.
• Blanc (2010) showed that it topologically simple for the Zariski topology.^[a]
• For the finite subgroups of the Cremona group see Dolgachev & Iskovskikh (2009).
• Zimmermann (2018) computed the abelianization of ${\displaystyle Cr_{2}(\mathbb {R} )}$. From this, she deduces that there is no analogue of Noether–Castelnuovo theorem in this context.^[6]

The Cremona group in higher dimensions

There is little known about the structure of the Cremona group in three dimensions and higher though many elements of it have been described.

There is no easy analogue of the Noether–Castelnouvo theorem, as Hudson (1927) showed that the Cremona group in dimension at least 3 is not generated by its elements of degree bounded by any fixed integer.

Blanc (2010) showed that it is (linearly) connected, answering a question of Serre (2010). Later, Blanc & Zimmermann (2018) showed that for any infinite field ${\displaystyle k}$, the group ${\displaystyle Cr_{n}(k)}$ is topologically simple^[a] for the Zariski topology, and even for the euclidean topology when ${\displaystyle k}$ is a local field.

Blanc, Lamy & Zimmermann (2021) proved that when ${\displaystyle k}$ is a subfield of the complex numbers and ${\displaystyle n\geq 3}$, then ${\displaystyle Cr_{n}(k)}$ is a simple group.

De Jonquières groups

A De Jonquières group is a subgroup of a Cremona group of the following form.^[7] Pick a transcendence basis ${\displaystyle x_{1},...,x_{n}}$ for a field extension of ${\displaystyle k}$. Then a De Jonquières group is the subgroup of automorphisms of ${\displaystyle k(x_{1},...,x_{n})}$ mapping the subfield ${\displaystyle k(x_{1},...,x_{r})}$ into itself for some ${\displaystyle r\leq n}$. It has a normal subgroup given by the Cremona group of automorphisms of ${\displaystyle k(x_{1},...,x_{n})}$ over the field ${\displaystyle k(x_{1},...,x_{r})}$, and the quotient group is the Cremona group of ${\displaystyle k(x_{1},...,x_{r})}$ over the field ${\displaystyle k}$. It can also be regarded as the group of birational automorphisms of the fiber bundle ${\displaystyle \mathbb {P} ^{r}\times \mathbb {P} ^{n-r}\to \mathbb {P} ^{r}}$.

When ${\displaystyle n=2}$ and ${\displaystyle r=1}$ the De Jonquières group is the group of Cremona transformations fixing a pencil of lines through a given point, and is the semidirect product of ${\displaystyle \mathrm {PGL} _{2}(k)}$ and ${\displaystyle \mathrm {PGL} _{2}(k(t))}$.