Search results for " surface"
showing 10 items of 2838 documents
Hodge Theory and Algebraic Cycles
2006
Algebraic cycles and Hodge theory, in particular Chow groups, Deligne cohomology and the study of cycle class maps were some of the themes of the Schwerpunkt ”Globale Methoden in der Komplexen Geometrie”. In this survey we report about several projects around the structure of (higher) Chow groups CH(X,n) [3] which the author has studied with his coauthors during this time by using different methods. In my opinion there are two interesting view points: first the internal structure of higher Chow groups, i.e., the existence of interesting elements and nontriviality of parts of their Bloch-Beilinson filtrations. This case has arithmetic and geometric features, and the groups in question show d…
Singularities of lightlike hypersurfaces in Minkowski four-space
2006
We classify singularities of lightlike hypersurfaces in Minkowski 4-space via the contact invariants for the corresponding spacelike surfaces and lightcones.
Transcendental lattices of some K 3-surfaces
2008
In a previous paper, (S2), we described six families of K3-surfaces with Picard- number 19, and we identified surfaces with Picard-number 20. In these notes we classify some of the surfaces by computing their transcendental lattices. Moreover we show that the surfaces with Picard-number 19 are birational to a Kummer surface which is the quotient of a non-product type abelian surface by an involution.
On a Theorem of Greuel and Steenbrink
2017
A famous theorem of Greuel and Steenbrink states that the first Betti number of the Milnor fibre of a smoothing of a normal surface singularity vanishes. In this paper we prove a general theorem on the first Betti number of a smoothing that implies an analogous result for weakly normal singularities.
Global 1-Forms and Vector Fields
2014
In this chapter we recall some fundamental facts concerning holomorphic 1-forms on compact surfaces: Albanese morphism, Castelnuovo–de Franchis Lemma, Bogomolov Lemma. We also discuss the logarithmic case, which is extremely useful in the study of foliations with an invariant curve. Finally we recall the classification of holomorphic vector fields on compact surfaces. All of this is very classical and can be found, for instance, in [2, Chapter IV] and 24, 35].
The Rationality Criterion
2014
In this chapter we explain a remarkable theorem of Miyaoka [32] which asserts that a foliation whose cotangent bundle is not pseudoeffective is a foliation by rational curves. The original Miyaoka’s proof can be thought as a foliated version of Mori’s technique of construction of rational curves by deformations of morphisms in positive characteristic [33].
Hypersurfaces of prescribed mean curvature over obstacles
1973
Let ~2 be a bounded domain in the euclidean space IR", n-> 2, with Lipschitz boundary ~ . We shall consider surfaces which are graphs of functions u defined on f2 having prescribed mean curvature H=H(x, u) with the side condition that they should be bounded from below by an obstacle ~b. The case H = 0 (minimal surfaces) has been discussed in detail by several authors, compare [6, 7, 12, 13, 17, 18, 20, 21, 24] of the references. Tomi [-31] has also investigated parametric surfaces with variable H. More general variational problems with obstructions have been discussed in [-9] and [-10]. During the session on "Variationsrechnung", held from June 18th to June 24th, 1972 in Oberwolfach, Mirand…
Dupin Cyclide Blends Between Quadric Surfaces for Shape Modeling
2004
We introduce a novel method to define Dupin cyclide blends between quadric primitives. Dupin cyclides are nonspherical algebraic surfaces discovered by French mathematician Pierre-Charles Dupin at the beginning of the 19th century. As a Dupin cyclide can be fully characterized by its principal circles, we have focussed our study on how to determine principal circles tangent to both quadrics being blended. This ensures that the Dupin cyclide we are constructing constitutes aG 1 blend. We use the Rational Quadratic Bezier Curve (RQBC) representation of circular arcs to model the principal circles, so the construction of each circle is reduced to the determination of the three control points o…
Dupin cyclide blends between non-natural quadrics of revolution and concrete shape modeling applications
2014
Abstract In this work, we focus on the blending of two quadrics of revolution by two patches of Dupin cyclides. We propose an algorithm for the blending of non-natural quadrics of revolution by decomposing the blending operation into two complementary sub-blendings, each of which is a Dupin cyclide-based blending between one of the two quadrics and a circular cylinder, thus enabling the direct computation of the two Dupin cyclide patches and offering better flexibility for shape composition. Our approach uses rational quadric Bezier curves to model the relevant arcs of the principal circles of Dupin cyclides. It is quite general and we have successfully used it for the blending of several n…
Minimal models of foliated algebraic surfaces
1999
MODELES MINIMAUX DES SURFACES ALGEBRIQUES FEUILLETEES. -Nous etudions les feuilletages holomorphes sur les surfaces algebriques du point de vue de la geometrie birationnelle. Apres avoir introduit une notion de modele minimal, nous classifions les feuilletages qui n'ont pas de modele minimal dans leur classe d'isomorphisme birationnel. Comme corollaire on obtient un resultat concernant la dynamique des diffeomorphismes polynomiaux.