The extension of regular and rational embeddings

Jelonek, Zbigniew

doi:10.1007/bf01457281

Cited by 58 publications

(31 citation statements)

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“…However, Craighero showed in [5] that for n ≥ 4 every embedding of k in k n is rectifiable. The same result was obtained by Jelonek in [13]. In fact Jelonek showed that if n ≥ 2r + 2, then every embedding of k r in k n is rectifiable, while Craighero showed this for all n ≥ 3r + 1.…”

Section: Triangular Derivations 1313supporting

confidence: 74%

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Triangular derivations related to problems on affine 𝑛-space

Essen

Rossum

2001

Proc. Amer. Math. Soc.

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Section: Triangular Derivations 1313supporting

confidence: 74%

“…The following easy argument, due to Jelonek in [13], shows that every embedding α : k r → k n is stably rectifiable, i.e., there exists m ≥ 1 such thatα :…”

Section: Triangular Derivations 1313mentioning

confidence: 99%

Triangular derivations related to problems on affine 𝑛-space

Essen

Rossum

2001

Proc. Amer. Math. Soc.

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“…It is natural to ask if there exists a Cremona transformation of P n that maps X 1 to X 2 , in this case we say that X 1 and X 2 are Cremona equivalent, see Definition 1.1 for the precise statement. This is somewhat related to the Abhyankar-Moh problem [1] and [11]. Quite surprisingly the main theorem in [19] states that this is the case as long as the codimension of X i is at least 2.…”

Section: Introductionmentioning

confidence: 99%

Equivalent birational embeddings II: divisors

Mella

Polastri

2011

Math. Z.

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Two divisors in $\P^n$ 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.Comment: v2 Exposition improved, thanks to referee, unconditional characterization of surfaces Cremona equivalent to a plan

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“…In particular, in [6], [7], [9] and [12] this problem was solved for the category of smooth complex algebraic affine varieties (where morphisms are polynomial mappings). The second aim of this paper is to generalize (and simplify) these results to the case of some other interesting pseudo-algebraic categories (see Definitions 4.3 and 4.11).…”

Section: Introductionmentioning

confidence: 99%

Manifolds with a unique embedding

Jelonek

2009

Colloq. Math.

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Abstract. We show that if X, Y are smooth, compact k-dimensional submanifolds of R n and 2k + 2 ≤ n, then each diffeomorphism φ : X → Y can be extended to a diffeomorphism Φ : R n → R n which is tame (to be defined in this paper). Moreover, if X, Y are real analytic manifolds and the mapping φ is analytic, then we can choose Φ to be also analytic.We extend this result to some interesting categories of closed (not necessarily compact) subsets of R n , namely, to the category of Nash submanifolds (with Nash, real-analytic and smooth morphisms) and to the category of closed semi-algebraic subsets of R n (with morphisms being semi-algebraic continuous mappings). In each case we assume that X, Y are k-dimensional and φ is an isomorphism, and under the same dimension restriction 2k + 2 ≤ n we assert that there exists an extension Φ : R n → R n which is an isomorphism and it is tame.The same is true in the category of smooth algebraic subvarieties of C n , with morphisms being holomorphic mappings and with morphisms being polynomial mappings.

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The extension of regular and rational embeddings

Cited by 58 publications

References 0 publications

Triangular derivations related to problems on affine 𝑛-space

Triangular derivations related to problems on affine 𝑛-space

Equivalent birational embeddings II: divisors

Manifolds with a unique embedding

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