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...

Given a real-valued continuous function defined on n-dimensional Euclidean space, we construct a Khalimsky-continuous integer-valued approximation. From a geometrical point of view, this digitization takes a hypersurface that is the graph of a function and produces a digital hypersurface—the graph of the digitized function.

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].