On a covering group theorem and its applications.
To any finite covering of degree between smooth complex projective manifolds, one associates a vector bundle of rank on whose total space contains . It is known that is ample when is a projective space ([Lazarsfeld 1980]), a Grassmannian ([Manivel 1997]), or a Lagrangian Grassmannian ([Kim Maniel 1999]). We show an analogous result when is a simple abelian variety and does not factor through any nontrivial isogeny . This result is obtained by showing that is -regular in the...
The classical Segre theory gives a necessary and sufficient condition for a plane curve to be a branch curve of a (generic) projection of a smooth surface in . We generalize this result for smooth surfaces in a projective space of any dimension in the following way: given two plane curves, and , we give a necessary and sufficient condition for to be the branch curve of a surface in and to be the image of the double curve of a -model of . In the classical Segre theory, a plane curve...
We study the geometrical properties of a unit vector field on a Riemannian 2-manifold, considering the field as a local imbedding of the manifold into its tangent sphere bundle with the Sasaki metric. For the case of constant curvature , we give a description of the totally geodesic unit vector fields for and and prove a non-existence result for . We also found a family of vector fields on the hyperbolic 2-plane of curvature which generate foliations on with leaves of constant intrinsic...