1,348 research outputs found
Real algebraic morphisms and Del Pezzo surfaces of degree 2
Let X and Y be affine nonsingular real algebraic varieties. A general problem in Real Algebraic Geometry is to try to decide when a smooth map f : X -> Y can be approximated by regular maps in the space of smooth mappings from X to Y, equipped with the compact-open topology. In this paper we give a complete solution to this problem when the target space is the usual 2-dimensional sphere and the source space is a geometrically rational real algebraic surface. The approximation result for real algebraic surfaces rational over R is due to J. Bochnak and W. Kucharz. Here we give a detailed description of the more interesting case, namely a real Del Pezzo surfaces of degree 2
Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle
Geometrization theorem, fibered case: Every three-manifold that fibers over
the circle admits a geometric decomposition. Double limit theorem: for any
sequence of quasi-Fuchsian groups whose controlling pair of conformal
structures tends toward a pair of projectively measured laminations that bind
the surface, there is a convergent subsequence. This preprint also analyzes the
quasi-isometric geometry of quasi-Fuchsian 3-manifolds. This eprint is based on
a 1986 preprint, which was refereed and accepted for publication, but which I
neglected to correct and return. The referee's corrections have now been
incorporated, but it is largely the same as the 1986 version (which was a
significant revision of a 1981 version).Comment: 32 pages, 6 figures, revision of 1986 preprin
Rationality problem of conic bundles
Let be a field with char , be an affine surface defined by
the equation where are separable
polynomials. We will investigate the rationality problem of in terms of the
polynomials and . The necessary and sufficient condition is with minor exceptions, where , (resp. ,
resp. ) being the number of such that and
(resp. and , resp.
and ). or according
to the behavior at . is a conic bundle over ,
whose rationality was studied by Iskovskikh. Iskovskikh formulated his results
in geometric language. This paper aims to give an algebraic counterpart.Comment: incorporates all of the content of arXiv:1308.090
An elementary approach to dessins d'enfants and the Grothendieck-Teichm\"uller group
We give an account of the theory of dessins d'enfants which is both
elementary and self-contained. We describe the equivalence of many categories
(graphs embedded nicely on surfaces, finite sets with certain permutations,
certain field extensions, and some classes of algebraic curves), some of which
are naturally endowed with an action of the absolute Galois group of the
rational field. We prove that the action is faithful. Eventually we prove that
this absolute Galois group embeds into the Grothendieck-Teichm\"uller group
introduced by Drinfel'd. There are explicit approximations of by
finite groups, and we hope to encourage computations in this area.
Our treatment includes a result which has not appeared in the literature yet:
the Galois action on the subset of regular dessins - that is, those exhibiting
maximal symmetry -- is also faithful.Comment: 58 pages, about 30 figures. Corrected a few typos. This version
should match the published paper in L'enseignement Mathematiqu
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