Cremona group

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

Historical origins

The Cremona group was introduced by the Italian mathematician Template:Harvs. 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. The mathematician Hilda Phoebe Hudson made contributions in the 1900s as well.

Basic properties

The Cremona group is naturally identified with the automorphism group ${\mathrm{A}\mathrm{u}\mathrm{t}}_k(k(x_1,\mathrm{...},x_n))$ of the field of the rational functions in $n$ indeterminates over $k$. Here, the field $k(x_1,\mathrm{...},x_n)$ is a pure transcendental extension of $k$, with transcendence degree $n$.

The projective general linear group ${\mathrm{P}\mathrm{G}\mathrm{L}}_{n+1}$ is contained in $C{r}_n$. The two are equal only when $n=0$ or $n=1$, in which case both the numerator and the denominator of a transformation must be linear.

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

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 $[x:y:z]\mapsto [yz:zx:xy]$, along with $\mathrm{P}\mathrm{G}\mathrm{L}(3,k)$, though there was some controversy about whether their proofs were correct. Template:Harvtxt 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.

• Template:Harvtxt showed that for an algebraically closed field $k$, the group $C{r}_2(k)$ is not simple.
• Template:Harvtxt showed that it is topologically simple for the Zariski topology.Template:Efn
• For the finite subgroups of the Cremona group see Template:Harvtxt.
• Template:Harvtxt computed the abelianization of $C{r}_2(ℝ)$. From this, she deduces that there is no analogue of Noether–Castelnuovo theorem in this context.

Geiser and Bertini involutions

The Geiser involution and Bertini involution are two of the classical non-linear involutions of the plane Cremona group. They arise from Del Pezzo surfaces of degree 2 and degree 1, respectively.

A Geiser involution is obtained by blowing up seven points of ${\mathbb{P}}^2$ in general position, producing a Del Pezzo surface $S$ of degree 2. The anticanonical linear system $|-K_S|$ defines a double cover $S\to {\mathbb{P}}^2$ branched over a smooth plane quartic, and the deck transformation is the Geiser involution. Via the blow-down $S\to {\mathbb{P}}^2$, this becomes a birational involution of the plane. Classically, if $p_1,\dots ,p_7$ are the seven base points, then a general point $x$ is sent to the ninth base point of the pencil of cubic curves through $p_1,\dots ,p_7,x$. The resulting Cremona transformation has degree 8.

Similarly, blowing up eight points of ${\mathbb{P}}^2$ in general position produces a Del Pezzo surface of degree 1. The linear system $|-2K_S|$ defines a double cover of a quadric cone in ${\mathbb{P}}^3$, and its deck transformation is the Bertini involution. In classical plane terms, if $p_1,\dots ,p_8$ are the base points, then for a general point $x$ one considers the net of sextics through $p_1,\dots ,p_8,x$ that are singular at $p_1,\dots ,p_8$; the Bertini involution sends $x$ to the fixed point of this net. Its degree is 17.

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 $\mathrm{B}\mathrm{i}\mathrm{r}({\mathbb{P}}^2)$ is conjugate to exactly one of three types: a de Jonquières involution, a Geiser involution, or a Bertini involution. 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. 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.

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 Template:Harvtxt showed that the Cremona group in dimension at least 3 is not generated by its elements of degree bounded by any fixed integer.

Template:Harvtxt showed that it is (linearly) connected, answering a question of Template:Harvtxt. Later, Template:Harvtxt showed that for any infinite field $k$, the group $C{r}_n(k)$ is topologically simpleTemplate:Efn for the Zariski topology, and even for the euclidean topology when $k$ is a local field.

Template:Harvtxt proved that when $k$ is a subfield of the complex numbers and $n\ge 3$, then $C{r}_n(k)$ is not a simple group.

De Jonquières groups

A De Jonquières group is a subgroup of a Cremona group of the following form. Pick a transcendence basis $x_1,\mathrm{...},x_n$ for a field extension of $k$. Then a De Jonquières group is the subgroup of automorphisms of $k(x_1,\mathrm{...},x_n)$ mapping the subfield $k(x_1,\mathrm{...},x_r)$ into itself for some $r\le n$. It has a normal subgroup given by the Cremona group of automorphisms of $k(x_1,\mathrm{...},x_n)$ over the field $k(x_1,\mathrm{...},x_r)$, and the quotient group is the Cremona group of $k(x_1,\mathrm{...},x_r)$ over the field $k$. It can also be regarded as the group of birational automorphisms of the fiber bundle ${\mathbb{P}}^r\times {\mathbb{P}}^{n-r}\to {\mathbb{P}}^r$.

When $n=2$ and $r=1$ 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 ${\mathrm{P}\mathrm{G}\mathrm{L}}_2(k)$ and ${\mathrm{P}\mathrm{G}\mathrm{L}}_2(k(t))$.

See also

Bibliography

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