124 research outputs found
Birationality of \'etale morphisms via surgery
We use a counting argument and surgery theory to show that if is a
sufficiently general algebraic hypersurface in , then any local
diffeomorphism of simply connected manifolds which is a
-sheeted cover away from has degree or (however all
degrees are possible if fails to be a local diffeomorphism at even
a single point). In particular, any \'etale morphism of
algebraic varieties which covers away from such a hypersurface must be
birational.Comment: 17 pages. Replaced to add further references and make language more
consistent with the literatur
Polynomial and regular images of R-n
We obtain new necessary conditions for an n-dimensional semialgebraic subset of R-n to be a polynomial image of R-n. Moreover, we prove that a large family of planar bidimensional semialgebraic sets with piecewise linear boundary are images of polynomial or regular maps, and we estimate in both cases the dimension of their generic fibers
Equivalent birational embeddings II: divisors
Two divisors in 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
Affine modifications and affine hypersurfaces with a very transitive automorphism group
We study a kind of modification of an affine domain which produces another
affine domain. First appeared in passing in the basic paper of O. Zariski
(1942), it was further considered by E.D. Davis (1967). The first named author
applied its geometric counterpart to construct contractible smooth affine
varieties non-isomorphic to Euclidean spaces. Here we provide certain
conditions which guarantee preservation of the topology under a modification.
As an application, we show that the group of biregular automorphisms of the
affine hypersurface given by the equation
where acts transitively on the
smooth part reg of for any We present examples of such
hypersurfaces diffeomorphic to Euclidean spaces.Comment: 39 Pages, LaTeX; a revised version with minor changes and correction
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