The group of Cremona transformations generated by linear maps and the standard involution

Blanc, Jérémy; Hedén, Isac

doi:10.5802/aif.2999

Cited by 14 publications

(19 citation statements)

References 14 publications

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“…In [BH14] it is shown that f A is contained in PGL n+1 (C), σ n , which implies that ψ 1 (f A ) preserves the vertical fibration and that its action on P n is the standard action.…”

Section: The Casementioning

confidence: 97%

“…Let f A be the birational transformation corresponding to A. In [BH14] it is shown that f A is contained in PGL n+1 (C), σ n .…”

Section: Appendixmentioning

confidence: 99%

“…In [BH14] it is shown that f 2 B is contained in PGL n+1 (C), σ n . By what we proved above, this gives…”

Section: Appendixmentioning

confidence: 99%

“…Blanc and Hedén studied the subgroup G n of Cr n generated by PGL n+1 (C) and the element σ n := [x −1 0 : · · · : x −1 n ] ( [BH14]). In particular, they show that G n is strictly contained in H n if and only if n is odd.…”

mentioning

confidence: 99%

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On homomorphisms between Cremona groups

Urech¹

2018

Ann. inst. Fourier

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We look at algebraic embeddings of the Cremona group in n variables Crn(C) to the groups of birational transformations Bir(M ) of an algebraic variety M . First we study geometrical properties of an example of an embedding of Cr 2 (C) into Cr 5 (C) that is due to Gizatullin. In a second part, we give a full classification of all algebraic embeddings of Cr 2 (C) into Bir(M ), where dim(M ) = 3 and generalize this result partially to algebraic embeddings of Crn(C) into Bir(M ), where dim(M ) = n + 1, for arbitrary n. In particular, this yields a classification of all algebraic PGL n+1 (C)-actions on smooth projective varieties of dimension n + 1 that can be extended to rational actions of Crn(C).Example 1.1. Assume that a variety M is birationally equivalent to P n ×N for some variety N . The standard action on the first factor yields an injective homomorphism of Cr n into Bir(P n ×N ) and therefore also into Bir(M ). We call embeddings of this type standard embeddings. In particular, we obtain in that way for all nonnegative integers m an injective homomorphism Cr n → Bir(P n × P m ).Example 1.2. A variety M is called stably rational if there exists a n such that M × P n is rational. There exist varieties of dimension larger than or equal to 3 that are stably rational but not rational (see [BCTSSD85]). We will see that two standard embeddings f 1 : Cr n → Bir(P n ×N ) and f 2 : Cr n → Bir(P n ×M ) are conjugate if and only if N and M are birationally equivalent (Lemma 3.3). So every class of birationally equivalent stably rational varieties of dimension k defines a different conjugacy class of injective homomorphisms Cr n → Bir(P m ) for m = n + k.

show abstract

“…In [BH14] it is shown that f A is contained in PGL n+1 (C), σ n , which implies that ψ 1 (f A ) preserves the vertical fibration and that its action on P n is the standard action.…”

Section: The Casementioning

confidence: 97%

“…Let f A be the birational transformation corresponding to A. In [BH14] it is shown that f A is contained in PGL n+1 (C), σ n .…”

Section: Appendixmentioning

confidence: 99%

“…In [BH14] it is shown that f 2 B is contained in PGL n+1 (C), σ n . By what we proved above, this gives…”

Section: Appendixmentioning

confidence: 99%

mentioning

confidence: 99%

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On homomorphisms between Cremona groups

Urech¹

2018

Ann. inst. Fourier

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“…One of the interesting subsets of the large group Bir(P n ) is the set H n of all f ∈ Bir(P n ) which only contracts rational hypersurfaces. It is known that G n ⊂ H n (See [BH14,§1]). On the other direction, [BH14] proved that G n = H n when n ≥ 3 is odd over any field k, by giving examples of monomial birational maps which only contract rational hypersurfaces but not in G n when n is odd.…”

mentioning

confidence: 99%