In birational geometry, the Cremona group, named after Luigi Cremona, is the group of birational automorphisms-dimensional projective space over a field , also known as Cremona transformations.

Historical origins

The Cremona group was introduced by the Italian mathematician . 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 of the field of the rational functions indeterminates is a pure transcendental extension, with transcendence degree.

The projective general linear group, 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, though there was some controversy about whether their proofs were correct. 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.

• showed that for an algebraically closed is not simple.

• showed that it is topologically simple for the Zariski topology.

• For the finite subgroups of the Cremona group see .

• computed the abelianization. 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.

branched over a smooth plane quartic, and the deck transformation is the Geiser involution.. The resulting Cremona transformation has degree 8.

defines a double cover of a quadric cone, and its deck transformation is the Bertini involution. to the fixed point of this net. Its degree is 17.

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

showed that it is (linearly) connected, answering a question of . Later, is topologically simple for the Zariski topology, and even for the euclidean topology is a local field.

is a subfield of the complex numbers 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 for a field extension. It has a normal subgroup, and the quotient group. It can also be regarded as the group of birational automorphisms of the fiber bundle.

the De Jonquières group is the group of Cremona transformations fixing a pencil of lines through a given point, and is the semidirect product.

See also

References

Notes

Bibliography

• ; reprinted 2012,