243 research outputs found

### Every connected sum of lens spaces is a real component of a uniruled algebraic variety

Let M be a connected sum of finitely many lens spaces, and let N be a
connected sum of finitely many copies of S^1xS^2. We show that there is a
uniruled algebraic variety X such that the connected sum M#N of M and N is
diffeomorphic to a connected component of the set of real points X(R) of X. In
particular, any finite connected sum of lens spaces is diffeomorphic to a real
component of a uniruled algebraic variety.Comment: Nouvelle version avec deux figure

### The group of automorphisms of a real rational surface is n-transitive

Let X be a rational nonsingular compact connected real algebraic surface.
Denote by Aut(X) the group of real algebraic automorphisms of X. We show that
the group Aut(X) acts n-transitively on X, for all natural integers n. As an
application we give a new and simpler proof of the fact that two rational
nonsingular compact connected real algebraic surfaces are isomorphic if and
only if they are homeomorphic as topological surfaces.Comment: Title changed, exposition improve

### Every orientable Seifert 3-manifold is a real component of a uniruled algebraic variety

AbstractWe show that any orientable Seifert 3-manifold is diffeomorphic to a connected component of the set of real points of a uniruled real algebraic variety, and prove a conjecture of JÃ¡nos KollÃ¡r

### Pencils on real curves

We consider coverings of real algebraic curves to real rational algebraic
curves. We show the existence of such coverings having prescribed topological
degree on the real locus. From those existence results we prove some results on
Brill-Noether Theory for pencils on real curves. For coverings having
topological degree 0 we introduce the covering number k and we prove the
existence of coverings of degree 4 with prescribed covering number.Comment: 27 pages, 12 figure

### Principal bundles over a smooth real projective curve of genus zero

Let H0 denote the kernel of the endomorphism, defined by z → (z/z-)2, of the real algebraic group given by the Weil restriction of C*. Let X be a nondegenerate anisotropic conic in P2R. The principal C*-bundle over the complexification XC, defined by the ample generator of Pic(XC), gives a principal H0-bundle FH0 over X through a reduction of structure group. Given any principal G-bundle EG over X, where G is any connected reductive linear algebraic group defined over R, we prove that there is a homomorphism ρ : H0 → G such that EG is isomorphic to the principal G-bundle obtained by extending the structure group of FH0 using ρ. The tautological line bundle over the real projective line P1 R, and the principal Z/2Z- bundle P1 C → P1 R, together give a principal Gm × (Z/2Z)-bundle F on P1 R. Given any principal G-bundle EG over P1R, where G is any connected reductive linear algebraic group defined over R, we prove that there is a homomorphism ρ : Gm × (Z/2Z) → G such that EG is isomorphic to the principal G-bundle obtained by extending the structure group of F using ρ

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