Cremona group

In algebraic geometry, the Cremona group, introduced by Cremona (1863, 1865), is the group of birational automorphisms of the ${\displaystyle n}$-dimensional projective space over a field ${\displaystyle k}$. It is denoted by ${\displaystyle Cr(\mathbb {P} ^{n}(k))}$ or ${\displaystyle Bir(\mathbb {P} ^{n}(k))}$ or ${\displaystyle Cr_{n}(k)}$.

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 ${\displaystyle k}$, or in other words a pure transcendental extension of ${\displaystyle k}$, with transcendence degree ${\displaystyle n}$.

The projective general linear group of order ${\displaystyle n+1}$, of projective transformations, is contained in the Cremona group of order ${\displaystyle n}$. 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.