The Cremona group of the plane is compactly presented

The Cremona group of the plane is compactly presented Susanna Zimmermann

This article shows that the Cremona group of the plane is compactly presented. To do this, we prove that it is a generalized amalgamated product of three of its algebraic subgroups (automorphisms of the plane and Hirzebruch surfaces) divided by one relation.

Combinatorics of Cremona monomial maps

One studies Cremona monomial maps by combinatorial means. Among the results is a simple integer matrix theoretic proof that the inverse of a Cremona monomial map is also defined by monomials of fixed degree, and moreover, the set of monomials defining the inverse can be obtained explicitly in terms of the initial data. A neat consequence is drawn for the plane Cremona monomial group, in particu...

Essentials of the Method of Maximal Singularities

We start with a very brief description of the principal events in the history of our subject. About 1870 M.Noether discovered [N] that the group of birational automorphisms of the projective plane BirP, which is known also as the Cremona group CrP, is generated by its subgroup AutP = AutC/C and any quadratic Cremona transformation τ , which in a certain system of homogeneous coordinates can be ...

Equivalent Birational Embeddings Ii: Divisors

Two divisors in Pn are said to be Cremona equivalent if there is a Cremona modification sending one to the other. We produce infinitely many non equivalent divisorial embeddings of any variety of dimension at most 14. Then we study the special case of plane curves and rational hypersurfaces. For the latter we characterise surfaces Cremona equivalent to a plane, under a mild assumption. Introduc...

Finite abelian subgroups of the Cremona group of the plane