# On Plane Cremona Transformations of Fixed Degree

@article{Bisi2012OnPC, title={On Plane Cremona Transformations of Fixed Degree}, author={Cinzia Bisi and Alberto Calabri and Massimiliano Mella}, journal={The Journal of Geometric Analysis}, year={2012}, volume={25}, pages={1108-1131} }

We study the quasi-projective variety $\operatorname{Bir}_{d}$ of plane Cremona transformations defined by three polynomials of fixed degree d and its subvariety $\operatorname{Bir}_{d}^{\circ}$ where the three polynomials have no common factor. We compute their dimension and the decomposition in irreducible components. We prove that $\operatorname{Bir}_{d}$ is connected for each d and $\operatorname{Bir}_{d}^{\circ}$ is connected when d<7.

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#### References

SHOWING 1-10 OF 39 REFERENCES

Plane polynomial automorphisms of fixed multidegree

- Mathematics
- 2008

Let $${\mathcal{G}}$$ be the group of polynomial automorphisms of the complex affine plane. On one hand, $${\mathcal {G}}$$ can be endowed with the structure of an infinite dimensional algebraic… Expand

Normal subgroups in the Cremona group

- Mathematics
- 2010

Let k be an algebraically closed field. We show that the Cremona group of all birational transformations of the projective plane $$ \mathbb{P}_{\mathbf{k}}^2 $$ is not a simple group. The strategy… Expand

On the length of polynomial automorphisms of the affine plane

- Mathematics
- 2002

Abstract. The automorphism group of the affine plane is a mysterious and challenging object. Although we know that it is an amalgamated product of two well known subgroups, many questions are still… Expand

On the Variety of Automorphisms of the Affine Plane

- Mathematics
- 1997

Abstract The main subject of our study is GA 2, n , the variety of automophisms of the affine plane of degree bounded by a positive integer n . After detailing some definitions and notations in… Expand

On the symmetric product of a rational surface

- Mathematics
- 1969

In his work on rational equivalence [5] Severi often raised this question: if the points of a nonsingular algebraic variety V are all rationally equivalent to each other, is V a unirational variety?… Expand

A plane foliation of degree different from 1 is determined by its singular scheme

- Mathematics
- 1999

Abstract We prove that a foliation 877-1 of degree ≠ 1 on P 2 is completely determined by its singular subscheme SingS( 877-2 ) of P 2 . We apply this result to obtain a similar characterization of… Expand

Defining Relations for the Cremona Group of the Plane

- Mathematics
- 1983

By methods of the geometry of rational surfaces and the topology of graphs and cell complexes connected with them, the author establishes defining relations, connecting projectives, and quadratic… Expand

Some families of polynomial automorphisms III

- Mathematics
- 2004

Abstract We prove that the closure (for the Zariski topology) of the set of polynomial automorphisms of the complex affine plane whose polydegree is ( c d − 1 , b , a ) contains all automorphisms of… Expand

Topologies and structures of the Cremona groups

- Mathematics
- 2013

We study the algebraic structure of the n-dimensional Cremona group and show that it is not an algebraic group of innite dimension (ind-group) if n 2. We describe the obstruction to this, which is of… Expand

Special subschemes of the scheme of singularities of a plane foliation

- Mathematics
- 2007

From the fact that a foliation by curves of degree greater than one, with isolated singularities, in the complex projective plane P2 is uniquely determined by its subscheme of singular points (the… Expand