When n satisfies an inequality which is almost best possible, we prove that the k-plane sections of every smooth, degree d, complex hypersurface in Pn dominate the moduli space of degree d hypersurfaces in Pk. As a corollary we prove that, for n sufficiently large, every smooth, degree d hypersurface in Pn satisfies a version of “rational simple connectedness”. 1. Statement of results In their ...

This note is an extension of the paper [HMP]. The main theorem of [HMP] states that a relatively smooth hypersurface (i.e. a hypersurface whose singular locus has sufficient large codimension with respect to its degree) is unirational. A mild modification can generalize this to complete intersections. As an application, we will show the Fano variety of a relatively smooth hypersurface is also u...

In our previous work (2002), we proved that an essential second-order hypersurface in an infinite-dimensional locally affine Riemannian Banach manifold is a Riemannian manifold of constant nonzero curvature. In this note, we prove the converse; in other words, we prove that a hypersurface of constant nonzero Riemannian curvature in a locally affine (flat) semiRiemannian Banach space is an essen...

A hypersurface of order (n+ 1)(ph − 1) in projective space of dimension n of prime characteristic p has an invariant monomial. This implies that a hypersurface of order (n+1)(ph−1)−1 determines an invariant point. A hypersurface of order d < n+ 1 in a projective space of dimension n of characteristic two has an invariant set of subspaces of dimension d− 1 determined by one linear condition on t...

In this article, we prove that there exists at least one chord orbit of Hamilton vector field connecting the Legendre submanifold on a given energy stable hypersurface in some symplectic manifolds. Démonstration de existence de chord orbital de hamiltonien en une stable hypersurface dans une variété symplectique. Résumé. Soit Σ une hypersurface close en variété symplectique, Il est de montrer q...

We study GCR-lightlike submanifolds of semi-Riemannian product manifolds. give some equivalent conditions for the integrability various distributions manifolds and investigate geometry leaves distributions.