Cremona group

The Cremona group was introduced by the Italian mathematician Luigi Cremona (1863, 1865).^[1] In retrospect, the British mathematician Isaac Newton is considered a founder of "the theory of Cremona transformations", having developed his "organic construction" to perform birational maps of the projective plane and applied them to resolve curve singularities, nearly two centuries before Cremona.^[2][3] The mathematician Hilda Phoebe Hudson made contributions in the 1900s as well.^[4]

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]

In two dimensions, Max Noether and Guido Castelnuovo showed that the complex Cremona group is generated by the standard quadratic transformation ${\displaystyle [x:y:z]\mapsto [yz:zx:xy]}$, 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 algebraically closed field ${\displaystyle k}$, the group ${\displaystyle Cr_{2}(k)}$ is not simple.
• Blanc (2010) showed that it is 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 Geiser involution and Bertini involution are two of the classical non-linear involutions of the plane Cremona group.^[7][8] They arise from Del Pezzo surfaces of degree 2 and degree 1, respectively.

A Geiser involution is obtained by blowing up seven points of ${\displaystyle \mathbb {P} ^{2}}$ in general position, producing a Del Pezzo surface ${\displaystyle S}$ of degree 2. The anticanonical linear system ${\displaystyle |-K_{S}|}$ defines a double cover ${\displaystyle S\to \mathbb {P} ^{2}}$ branched over a smooth plane quartic, and the deck transformation is the Geiser involution.^[9][10] Via the blow-down ${\displaystyle S\to \mathbb {P} ^{2}}$, this becomes a birational involution of the plane. Classically, if ${\displaystyle p_{1},\dots ,p_{7}}$ are the seven base points, then a general point ${\displaystyle x}$ is sent to the ninth base point of the pencil of cubic curves through ${\displaystyle p_{1},\dots ,p_{7},x}$.^[7][11] The resulting Cremona transformation has degree 8.^[12]

Similarly, blowing up eight points of ${\displaystyle \mathbb {P} ^{2}}$ in general position produces a Del Pezzo surface of degree 1. The linear system ${\displaystyle |-2K_{S}|}$ defines a double cover of a quadric cone in ${\displaystyle \mathbb {P} ^{3}}$, and its deck transformation is the Bertini involution.^[13][14] In classical plane terms, if ${\displaystyle p_{1},\dots ,p_{8}}$ are the base points, then for a general point ${\displaystyle x}$ one considers the net of sextics through ${\displaystyle p_{1},\dots ,p_{8},x}$ that are singular at ${\displaystyle p_{1},\dots ,p_{8}}$; the Bertini involution sends ${\displaystyle x}$ to the fixed point of this net.^[7] Its degree is 17.^[15]

Over an algebraically closed field of characteristic different from 2, Bayle and Beauville's modern proof of the classical theorem on birational involutions of the plane shows that every non-trivial involution in ${\displaystyle \mathrm {Bir} (\mathbb {P} ^{2})}$ is conjugate to exactly one of three types: a de Jonquières involution, a Geiser involution, or a Bertini involution.^[7] The normalized fixed curve of a Geiser involution is a non-hyperelliptic curve of genus 3, while that of a Bertini involution is a non-hyperelliptic curve of genus 4 whose canonical model lies on a singular quadric.^[7] Consequently, conjugacy classes of Geiser involutions are parametrized by isomorphism classes of non-hyperelliptic genus-3 curves, and conjugacy classes of Bertini involutions by isomorphism classes of non-hyperelliptic genus-4 curves whose canonical model lies on a singular quadric.^[7]

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 not a simple group.

A De Jonquières group is a subgroup of a Cremona group of the following form.^[16] 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 ${\displaystyle \mathrm {PGL} _{2}(k(t))}$.