Elements and cyclic subgroups of finite order of the Cremona group

Blanc, Jérémy

doi:10.4171/cmh/231

Cited by 29 publications

(62 citation statements)

References 19 publications

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“…Since C p is a twisted cubic, it is an OADP variety in its span C p P 3 and Φ f induces a Cremona involution Φ Cp : C p C p . Since Φ Cp is an involution of bidegree (3,3) in P 3 , we deduce that Φ f is of the same type, concluding the proof.…”

Section: Annales De L'institut Fouriermentioning

confidence: 57%

“…By definition, a cubo-cubic Cremona transformation is a birational map of a projective space of bidegree (3,3). Such maps have been studied in low dimension (see [17,27] for instance) but except in dimension 2 and 3 no general classification or structure results are known for them.…”

Section: From Quadro-quadric Cremona Transformations To Cubo-cubic Onesmentioning

confidence: 99%

“…(9) Note however that the description of the linear system of type tan [3] (//) is not correct in [28]. (10) To recover exactly the linear system x 2 , xy, xz, y 2 − xt, 2yz − xu, z 2 appearing in Table 8, one has to take S = V (y 2 − xt, 2zt − yu, 2yz − xu) and = V (x, y, z).…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

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Quadro-quadric Cremona transformations in low dimensions via the JC-correspondence

Pirio¹,

Russo²

2014

Ann. inst. Fourier

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We apply the results of [29] to study quadro-quadric Cremona transformations in low-dimensional projective spaces. In particular we describe new very simple families of such birational maps and obtain complete and explicit classifications in dimension four and five. † Partially supported by the franco-italian research network GRIFGA and by the P.R.A. of the Università degli Studi di Catania. 1 Two Cremona maps f 1 , f 2 of P n are said to be linearly equivalent if f 2 = g • f 1 • h for some linear automorphisms g, h ∈ Aut(P n ). 2 Note that Jordan algebras are considered up to isomorphisms in the present paper whereas they were originally considered up to isotopies in [29]. If the later setting is more natural from a categorical point of view (see [29, Remark 4.2]), the XJC-correspondence still holds true at the set-theoretical level when considering rank 3 Jordan algebras up to isomorphisms. The reason behind this is the well-known fact that over the field of complex numbers, two isotopic Jordan algebras are actually isomorphic (cf. [21, Problem 7.2.(6)]). 1 1. NOTATION AND DEFINITIONS 1.0.1. General notation and definitions. If f = [f 0 : • • • : f n ] : P nP n is a Cremona transformation, then B f ⊂ P n will be the base locus scheme of f , that is the scheme defined by the n+1 homogeneous forms f 0 , . . . , f n that are assumed without common factor. If all these are of degree d 1 ≥ 1 and if the inverse f −1 : P n P n of f is defined by forms of degree d 2 without common factorwe say that f has bidegree (d 1 , d 2 ) and we put bdeg(f ) = (d 1 , d 2 ). The set of Cremona transformations of P n of bidegree (d 1 , d 2 ) will be indicated by Bir d1,d2 (P n ).By definition, a quadro-quadric Cremona transformation is just a Cremona transformation of bidegree (2, 2). Two Cremona transformations f, f ∈ Bir d1,d2 (P n ) are said to be linearly equivalent (or just equivalent for short) if there exist projective transformations ℓ 1 , ℓ 2 :We define the type of a Cremona transformation f : P n P n as the irreducible component of the Hilbert scheme of P n to which B f belongs. We put P i (t) = t+i i so that P 0 = 1, P 1 (t) = t + 1 and P 2 (t) = (t+2) (t+1)

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Section: Annales De L'institut Fouriermentioning

confidence: 57%

Section: From Quadro-quadric Cremona Transformations To Cubo-cubic Onesmentioning

confidence: 99%

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Quadro-quadric Cremona transformations in low dimensions via the JC-correspondence

Pirio¹,

Russo²

2014

Ann. inst. Fourier

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“…Even if the generators of Cr (2) have been known for a century now, many other properties of this group are still mysterious. Only after decades, see [26], a complete set of relations has been described, and more recently the non simplicity of Cr (2) has been showed, [9], and a good understanding of its finite subgroups has been achieved, see [17] and [5]. This brief and fairly incomplete list is only meant to stress the difficulties and the large unknown parts in the study of Cr (2), for a more complete picture the interested reader should refer to [15] and [14].…”

Section: Introductionmentioning

confidence: 99%

“…Nevertheless this group is not less mysterious and challenging than the entire Cremona group. Jung's description yields a natural decomposition Aut(C 2 ) = A ∪ G[2] ∪ G [3] ∪ G [2,2] ∪ G [4] ∪ G [5] · · · into sets of polynomial automorphisms of multidegree (d 1 , · · · , d m ) and the affine subgroup A. In [22], Friedland and Milnor proved that G[d 1 , · · · , d m ] is a smooth analytic manifold of dimension (d 1 +d 2 +· · · d m +6).…”

Section: Introductionmentioning

confidence: 99%

On Plane Cremona Transformations of Fixed Degree

2013

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IntroductionThe birational geometry of algebraic varieties is governed by the group of birational self-maps. It is in general very difficult to determine this group for an arbitrary variety and as a matter of facts only few examples are completely understood. The special case of the projective plane attracted lots of attention since the XIX th -century. The pioneering work of Cremona and then the classical geometers of the Italian and German school were able to give partial descriptions of it but it was only after Noether and Castelnuovo that generators of the group were described. The Noether-Castelnuovo Theorem, see for instance [1], states that the group of birational self-maps of P 2 , usually called the plane Cremona group and denoted by Cr (2), is generated by linear automorphisms of P 2 and a single birational non biregular map, the so-called elementary quadratic transformation σ : P 2 P 2 defined by σ([x : y : z]) = [yz : xz : xy]. Even if the generators of Cr(2) have been known for a century now, many other properties of this group are still mysterious. Only after decades, see [26], a complete set of relations has been described, and more recently the non simplicity of Cr(2) has been showed, [9], and a good understanding of its finite subgroups has been achieved, see [17] and [5]. This brief and fairly incomplete list is only meant to stress the difficulties and the large unknown parts in the study of Cr(2), for a more complete picture the interested reader should refer to [15] and [14]. Amid all its subgroups the one associated to polynomial automorphisms of the plane, Aut(C 2 ), attracted even more attention than Cr(2) itself, [3]. The generators of Aut(C 2 ) are known since 1942, [29], and later on, [32], has been proved that Aut(C 2 ) is the amalgamated product of two of its subgroups, more precisely of the affine and elementary ones. Nevertheless this group is not less mysterious and challenging than the entire Cremona group. Jung's description yields a natural decompositioninto sets of polynomial automorphisms of multidegree (d 1 , · · · , d m ) and the affine subgroup A. In [22], Friedland and Milnor proved that G[d 1 , · · · , d m ] is a smooth analytic manifold of dimension (d 1 +d 2 +· · · d m +6). Later on, Furter, [23], computed the number of irreducible components of polynomial automorphisms of C 2 with fixed degree less or equal to 9 and proved that the variety of polynomial automorphisms of the plane with degree bounded by the positive integer n, is reducible when n 4. Later on Edo and Furter,[18], studied some degenerations of the multidegrees: for example they were able to show that G[3] ∩ G[2, 2] = ∅ using the lower semicontinuity of the length of a plane polynomial automorphism as a word of the amalgamated product, [24]. Contrarily to what happens in the Cremona group Cr(2), see [6], the group of polynomial automorphisms G of the plane can be endowed with a structure of an infinite-dimensional algebraic group. Denoted by G d the set of polynomial automorphisms of fixed degree d, Furte...

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Embeddings of SL(2; ℤ) into the cremona group

Blanc

Déserti

2012

Transformation Groups

Self Cite

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Geometric and dynamic properties of embeddings of SL(2, Z) into the Cremona group are studied. Infinitely many nonconjugate embeddings that preserve the type (i.e., that send elliptic, parabolic and hyperbolic elements onto elements of the same type) are provided. The existence of infinitely many nonconjugate elliptic, parabolic and hyperbolic embeddings is also shown. In particular, a group G of automorphisms of a smooth surface S obtained by blowing up 10 points of the complex projective plane is given. The group G is isomorphic to SL(2, Z), preserves an elliptic curve and all its elements of infinite order are hyperbolic.

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Elements and cyclic subgroups of finite order of the Cremona group

Cited by 29 publications

References 19 publications

Quadro-quadric Cremona transformations in low dimensions via the JC-correspondence

Quadro-quadric Cremona transformations in low dimensions via the JC-correspondence

On Plane Cremona Transformations of Fixed Degree

Embeddings of SL(2; ℤ) into the cremona group

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