Abstract
The Cremona group is the group of birational transformations of the plane. A birational transformation for which there exists a pencil of lines which is sent onto another pencil of lines is called a Jonqui\`eres transformation. By the famous Noether-Castelnuovo theorem, every birational transformation $f$ is a product of Jonqui\`eres transformations. The minimal number of factors in such a product will be called the length, and written $\mathrm{lgth}(f)$. Even if this length is rather unpredictable, we provide an explicit algorithm to compute it, which only depends on the multiplicities of the linear system of $f$. As an application of this computation, we give a few properties of the dynamical length of $f$ defined as the limit of the sequence $n \mapsto \mathrm{lgth} (f^n) / n$. It follows for example that an element of the Cremona group is distorted if and only if it is algebraic. The computation of the length may also be applied to the so called Wright complex associated with the Cremona group: This has been done recently by Lonjou. Moreover, we show that the restriction of the length to the automorphism group of the affine plane is the classical length of this latter group (the length coming from its amalgamated structure). In another direction, we compute the lengths and dynamical lengths of all monomial transformations, and of some Halphen transformations. Finally, we show that the length is a lower semicontinuous map on the Cremona group endowed with its Zariski topology.
Highlights
We show that the restriction of the length to the automorphism group of the affine plane is the classical length of this latter group
It is not possible to have a family of birational maps with homaloidal type (8; 4, 35, 12) which degenerates to a birational map of homaloidal type (8; 43, 23, 13) as the first homaloidal type has length 2 and the second has length 3
When we want to decompose a birational transformation of P2, we have to study the multiplicities of the linear system at points, and the position of the points
Summary
Let us fix an algebraically closed field k. — Lemma 1.3 (2) asserts that any Cremona transformation f ∈ Bir(P2) admits finitely many predecessors up to right multiplication by an element of Aut(P2). Computing a sequence of predecessors (which is algorithmic, as said before, and whose homaloidal types are uniquely determined by the one of the map we start with) yields a finite algorithm to compute the length of any element of Bir(P2), as our main theorem states: Theorem 1.5. The lengths of all maps of some given degree can be computed (see Section 4.1 for tables up to degree 12) Another consequence of Theorem 1.5 is that the length of an element of Aut(A2), viewed as an element of Bir(P2), is the same as the classical length given by the amalgamated product structure (Proposition 4.2). We relate these notions with some decompositions of elements in GL2(Z) and with continued fractions
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