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Optimal degree construction of real algebraic plane nodal curves with prescribed topology. I. The orientable case.

Francisco Santos (1997)

Revista Matemática de la Universidad Complutense de Madrid

We study a constructive method to find an algebraic curve in the real projective plane with a (possibly singular) topological type given in advance. Our method works if the topological model T to be realized has only double singularities and gives an algebraic curve of degree 2N+2K, where N and K are the numbers of double points and connected components of T. This degree is optimal in the sense that for any choice of the numbers N and K there exist models which cannot be realized algebraically with...

Quadratic mappings and configuration spaces

Gia Giorgadze (2003)

Banach Center Publications

We discuss some approaches to the topological study of real quadratic mappings. Two effective methods of computing the Euler characteristics of fibers are presented which enable one to obtain comprehensive results for quadratic mappings with two-dimensional fibers. As an illustration we obtain a complete topological classification of configuration spaces of planar pentagons.

Real cubic hypersurfaces and group laws.

Johannes Huisman (2004)

Revista Matemática Complutense

Let X be a real cubic hypersurface in Pn. Let C be the pseudo-hyperplane of X, i.e., C is the irreducible global real analytic branch of the real analytic variety X(R) such that the homology class [C] is nonzero in Hn-1(Pn(R),Z/2Z). Let L be the set of real linear subspaces L of Pn of dimension n - 2 contained in X such that L(R) ⊆ C. We show that, under certain conditions on X, there is a group law on the set L. It is determined by L + L' + L = 0 in L if and only if there is a real hyperplane H...

Real hypersurfaces with many simple singularities.

Eric Westenberger (2005)

Revista Matemática Complutense

In this paper we present constructions of real hypersurfaces with many simple singularities and deduce an asymptotical optimal existence result for hypersurfaces corresponding to T-smooth germs of the equisingular stratum. We proceed along the lines of Shustin-Westenberge (2004) where analogous results were shown for the complex case.

Real Kodaira surfaces.

Paola Frediani (2004)

Collectanea Mathematica

In this paper we give the topological classification of real primary Kodaira surfaces and we describe in detail the structure of the corresponding moduli space. Moreover, we use the notion of the orbifold fundamental group of a real variety, which was also the main tool in the classification of real hyperelliptic surfaces achieved in [10]. Our first result is that if (S,sygma) is a real primary Kodaira surface, then the differentiable tupe of the pair (S,sygma) is completely determined by the orbifold...

Real singular Del Pezzo surfaces and 3-folds fibred by rational curves, II

Fabrizio Catanese, Frédéric Mangolte (2009)

Annales scientifiques de l'École Normale Supérieure

Let W → X be a real smooth projective 3-fold fibred by rational curves such that W ( ℝ ) is orientable. J. Kollár proved that a connected component N of W ( ℝ ) is essentially either Seifert fibred or a connected sum of lens spaces. Answering three questions of Kollár, we give sharp estimates on the number and the multiplicities of the Seifert fibres (resp. the number and the torsions of the lens spaces) when X is a geometrically rational surface. When N is Seifert fibred over a base orbifold F , our result generalizes...

Semi-algebraic neighborhoods of closed semi-algebraic sets

Nicolas Dutertre (2009)

Annales de l’institut Fourier

Given a closed (not necessarly compact) semi-algebraic set X in ℝ n , we construct a non-negative semi-algebraic 𝒞 2 function f such that X = f - 1 ( 0 ) and such that for δ > 0 sufficiently small, the inclusion of X in f - 1 ( [ 0 , δ ] ) is a retraction. As a corollary, we obtain several formulas for the Euler characteristic of  X .

Sur la première classe de Stiefel-Whitney de l’espace des applications stables réelles vers l’espace projectif

Nicolas Puignau (2010)

Annales de l’institut Fourier

L’espace de module des applications stables vers l’espace projectif possède naturellement une structure réelle dont la partie réelle est une variété projective normale. Cette dernière est un espace de module pour les courbes spatiales rationnelles réelles avec des points marqués réels. Puisque le lieu singulier est de codimension au moins deux, une première classe de Stiefel-Whitney est bien définie. Dans cet article nous déterminons un représentant pour la première classe de Stiefel-Whitney dans...

Sur la réalité des points doubles des courbes gauches

Daniel Pecker (1999)

Annales de l'institut Fourier

Une courbe réelle peut avoir des points doubles ordinaires de trois types différents : des points doubles réels à tangentes réelles, des points doubles réels isolés dans le domaine réel et des points doubles imaginaires. Soient α , β , γ , d ≥ n ≥ 2 des entiers tels que α + β + 2 γ ≤ C ( d , n ) (où C ( d , n ) désigne la borne de Castelnuovo). On construit une courbe réelle irréductible de degré d , non dégénérée dans l’espace projectif P n (i.e. non contenue dans un hyperplan) ayant pour seules singularités α points doubles réels à tangentes réelles,...

Teoría métrica de curvas semialgebráicas.

Lev Birbrair, Alexandre C. G. Fernandes (2000)

Revista Matemática Complutense

We present a complete bi-Lipschitz classification of germs of semialgebraic curves (semialgebraic sets of the dimension one). For this purpose we introduce the so-called Hölder Semicomplex, a bi-Lipschitz invariant. Hölder Semicomplex is the collection of all first exponents of Newton-Puiseux expansions, for all pairs of branches of a curve. We prove that two germs of curves are bi-Lipschitz equivalent if and only if the corresponding Hölder Semicomplexes are isomorphic. We also prove that any Hölder...

The number of conics tangent to five given conics: the real case.

Felice Ronga, Alberto Tognoli, Thierry Vust (1997)

Revista Matemática de la Universidad Complutense de Madrid

It is a classical result, first established by de Jonquières (1859), that generically the number of conics tangent to 5 given conics in the complex projective plane is 3264. We show here the existence of configurations of 5 real conics such that the number of real conics tangent to them is 3264.

Currently displaying 61 – 80 of 93