On degenerations of plane Cremona transformations

Blanc, Jérémy; Calabri, Alberto

doi:10.1007/s00209-015-1539-z

Cited by 6 publications

(8 citation statements)

References 19 publications

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“…In our text and by definition, each homaloidal type will be of this form, i.e. will be the homaloidal type of at least one birational transformation of P 2 (in [AC02,BC16a] such homaloidal types are called proper homaloidal types). We also define the comultiplicity of f to be deg(f ) − max i m i .…”

Section: ])mentioning

confidence: 99%

“…For instance, it is not possible to have a family of birational maps with homaloidal type (8; 4, 3 5 , 1 2 ) which degenerates to a birational map of homaloidal type (8; 4 3 , 2 3 , 1 3 ) as the first homaloidal type has length 2 and the second has length 3 (see Section 4.1). See [BCM15,BC16a] for more details on this question.…”

Section: Lower Semicontinuity Of the Lengthmentioning

confidence: 99%

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Length in the Cremona group

Blanc¹,

Furter²

2019

Annales Henri Lebesgue

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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 Jonquières transformation. By the famous Noether-Castelnuovo theorem, every birational transformation f is a product of Jonquières transformations. The minimal number of factors in such a product will be called the length, and written 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 → 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.

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Section: ])mentioning

confidence: 99%

Section: Lower Semicontinuity Of the Lengthmentioning

confidence: 99%

Length in the Cremona group

Blanc¹,

Furter²

2019

Annales Henri Lebesgue

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“…Thus, in all these cases one has µ 1 + 2 = d − 3 + 2 = d − 1. By [3,Proposition 5.2] this means that when the highest multiplicity is d − 3, except for case (1.1), the Cremona map belongs to the closure of the set of Cremona maps of degree d + 1.…”

Section: Highest Virtual Multiplicity D −mentioning

confidence: 99%

“…A rational map F : P n P n is defined by a linear system of n + 1 independent forms of the same degree in R. If F is birational then it is called a Cremona map. Beyond the notable modern development on the structure of the Cremona group, a body of results on the nature, structure and uses of an individual Cremona map has lately come up that draws on modern geometric and algebraic tools (for a very short sample, see [2], [3], [6], [4], [5], [9], [17], [18], [19], [21], [24], [25]). This paper goes along this line and focus only on the plane case (i.e., n = 2).…”

Section: Introductionmentioning

confidence: 99%

Homaloidal nets and ideals of fat points I

Ramos

Simis

2016

LMS J. Comput. Math.

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We consider plane Cremona maps with proper base points and the base ideal generated by the linear system of forms defining the map. The object of this work is to study the link between the algebraic properties of the base ideal and those of the ideal of these points fattened by the virtual multiplicities arising from the linear system. We reveal conditions which naturally regulate this association, with particular emphasis on the homological side. While most classical numerical inequalities concern the three highest virtual multiplicities, here we emphasize also the role of one single highest multiplicity. In this vein we describe classes of Cremona maps for large and small values of the highest virtual multiplicity. We also deal with the delicate question as to when is the base ideal non-saturated and consider the structure of its saturation.

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“…See the proof of Theorem 4.6 in [4] or the proof of Lemma 4.1 in [1] for other situations where this technique is used. Proof Changing φ to α −1 φ where α ∈ A is the affine part or φ, we can assume that the affine part of φ is id.…”

Section: Closed Subgroupmentioning

confidence: 99%