Pseudo Ricci symmetric real hypersurfaces of a complex projective space are classified and it is proved that there are no pseudo Ricci symmetric real hypersurfaces of the complex projective space CPn for which the vector field ξ from the almost contact metric structure (φ, ξ, η, g) is a principal curvature vector field.

A smooth, nondegenerate hypersurface in projective space contains no linear subvarieties of greater than half its dimension. It can contain linear subvarieties of half its dimension. This note proves that a smooth hypersurface of degree d ≥ 3 contains at most finitely many such subvarieties. Let k be a field. Let X ⊂ Pk be a hypersurface of degree d > 1. For each integer m > 0, denote by Fm(X) ...

This paper shows that the general hypersurface of degree ≥ 6 in projective four space cannot support an indecomposable rank two vector bundle which is Arithmetically CohenMacaulay and four generated. Equivalently, the equation of the hypersurface is not the Pfaffian of a four by four minimal skewsymmetric matrix.

Kobayshi’s conjecture proposes that the general hypersurface X of P3 of degree ≥ 5 is hyperbolic. This paper shows that nodal hypersurfaces X with many nodes are algebraically quasi-hyperbolic, i.e. there are only finitely many rational and elliptic curves on X. A key element is the existence of many symmetric log-differentials in the minimal resolution of the nodal hypersurface.

Given a globally hyperbolic spacetime, the existence of a (smooth) spacelike Cauchy hypersurface S has been proven recently. Here, we prove that any acausal spacelike compact submanifold with boundary can be smoothly extended to a spacelike Cauchy hypersurface. Apart from the interest as a purely geometric question (applicable to the Cauchy problem in General Relativity), the result is motivate...

We revisit known results about the Milnor class of a singular complex hypersurface, and rephrase some of them in a way that allows for a better comparison with the topological formula of Cappell and Shaneson for the Lclass of such a hypersurface. Our approach is based on Verdier’s specialization property for the Chern-MacPherson class, and simple constructible function calculus.

For a general hypersurface of degree d in projective n-space, if n ≥ d the spaces of 2-pointed rational curves on the hypersurface are rationally connected; thus the hypersurfaces are rationally simply connected. This paper proves stronger versions of theorems in [HS05].

Consider a compact embedded hypersurface ? t in R n+1 which moves with speed determined at each point by a function F (1 ; : : : ; n ; t) of its principal curvatures, for 0 t < T: We assume the problem is degenerate parabolic, that is, that F (; t) is nondecreasing in each of the principal curvatures 1 ; : : : ; n : We shall show that for t > 0 the hypersurface ? t sat-isses local a priori Lips...

We consider the problem of constructing roadmaps of real algebraic sets. This problem was introduced by Canny to answer connectivity questions and solve motion planning problems. Given s polynomial equations with rational coefficients, of degree D in n variables, Canny’s algorithm has a Monte Carlo cost of s log(s)D 2) operations in Q; a deterministic version runs in time s log(s)D 4). A subseq...

Gómez-Mont, Seade and Verjovsky introduced an index, now called GSV-index, generalizing the Poincaré-Hopf index to complex vector fields tangent to singular hypersurfaces. The GSV-index extends to the real case. This is a survey paper on the joint research with Gómez-Mont and Giraldo about calculating the GSV-index IndV±,0(X) of a real vector field X tangent to a singular hypersurface V = f(0)....