Cremona transformation

A birational transformation of a projective space $\def\P{\mathbb{P}} \P_k^n$, $n\ge 2$, over a field $k$. Birational transformations of the plane and of three-dimensional space were systematically studied (from 1863 on) by L. Cremona. The group of Cremona transformations is also named after him — the Cremona group, and is denoted by $\def\Cr{\rm{Cr}}\Cr(\P_k^n)$.

The simplest examples of Cremona transformations which are not projective transformations are quadratic birational transformations of the plane. In non-homogeneous coordinates $(x,y)$ they may be expressed as linear-fractional transformations $$x\mapsto \frac{a_1x+b_1y+c_1}{a_2x+b_2y+c_2}, \quad y\mapsto \frac{a_3x+b_3y+c_3}{a_4x+b_4y+c_4}.$$ Among these transformations, special consideration is given to the standard quadratic transformation $\tau$: $$(x,y)\mapsto \left( { \frac{1}{x},\frac{1}{y} } \right),$$ or, in homogeneous coordinates, $$(x_0,x_1,x_2) \mapsto(x_1x_2,x_0x_2,x_0x_1). $$ This transformation is an isomorphism off the coordinate axes: $$\tau:\P_k^2\setminus \{x_0x_1x_2 = 0\} \tilde\to \P_k^2 \setminus \{x_0x_1x_2 = 0 \}, $$ it has three fundamental points (points at which it is undefined) $(0,0,1)$, $(0,1,0)$ and $(1,0,0)$, and maps each coordinate axis onto the unique fundamental point not contained in that axis.

By Noether's theorem (see Cremona group), if $k$ is an algebraically closed field, each Cremona transformation of the plane $\P_k^2$ can be expressed as a composition of quadratic transformations.

An important place in the theory of Cremona transformations is occupied by certain special classes of transformations, in particular — Geiser involutions and Bertini involutions (see [1]). A Geiser involution $\alpha : \P_k^2 \to \P_k^2$ is defined by a linear system of curves of degree 8 on $\P_k^2$, which pass with multiplicity 3 through 7 points in general position. A Bertini involution $\beta : \P_k^2 \to \P_k^2$ is defined by a linear system of curves of degree 17 on $\P_k^2$, which pass with multiplicity 6 through 8 points in general position.

A Cremona transformation of the form $$x\mapsto x,$$

$$y\mapsto \frac{P(x)y+Q(x)}{R(x)y+S(x)},\quad P,Q,R,S\in k[x],$$ is called a de Jonquières transformation. De Jonquières transformations are most naturally interpreted as birational transformations of the quadric $\P_k^1\times \P_k^1$ which preserve projection onto one of the factors. One can then restate Noether's theorem as follows: The group ${\rm Bir}(P^1\times P^1)$ of birational automorphisms of the quadric is generated by an involution $\sigma$ and by the de Jonquières transformations, where $\sigma\in {\rm Aut}(P^1\times P^1)$ is the automorphism defined by permutation of factors.

Any biregular automorphism of the affine space $\def\A{\mathbb{A}}\A_k^n$ in $\P_k^n$ may be extended to a Cremona transformation of $\P_k^n$, so that ${\rm Aut}(\P^1\times \P^1) \subset {\rm Cr}(\P_k^n)$. When $n=2$ the group ${\rm Aut}(\A_k^2)$ is generated by the subgroup of affine transformations and the subgroup of transformations of the form $$x\mapsto ax+b,\quad y\mapsto cy+Q(x),$$

$$a\ne 0,\quad c\ne 0,\quad a,b\in k,\; Q(x)\in k[x],$$ moreover, it is the amalgamated product of these subgroups [5]. The structure of the group ${\rm Aut}(\A_k^n)$, $n\ge 3$, is not known. In general, up to the present time (1987) no significant results have been obtained concerning Cremona transformations for dimensions $n\ge 3$.

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The fact that ${\rm Aut}(\P_k^2)$ is the amalgamated product of the subgroup of affine transformations (cf. Affine transformation) with that of the transformations (*) was first proved (for ${\rm char}\; k = 0$) by H.W.E. Jung [a1]; the case of arbitrary ground field was proved by W. van der Kulk [a2].

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