Let X ⊂ P be a smooth irreducible projective threefold, and d its degree. In this paper we prove that there exists a constant β such that for all X containing a smooth ruled surface as hyperplane section and not contained in a fourfold of degree less than or equal to 15, d ≤ β. Under some more restrictive hypothesis we prove an analogous result for threefolds containing a smooth ruled surface a...

We explore several problems related to ruled polygons. Given a ruling of a polygon P , we consider the Reeb graph of P induced by the ruling. We define the Reeb complexity of P , which roughly equates to the minimum number of points necessary to support P . We give asymptotically tight bounds on the Reeb complexity that are also tight up to a small additive constant. When restricted to the set ...

We give a classification of the counterexamples of the Kawamata-Viehweg vanishing on a geometrically ruled surface in terms of the Tango invariant of the base curve.

A complex ruled surface admits an iterated blow-up encoded by a parabolic structure with rational weights. Under a condition of parabolic stability, one can construct a Kähler metric of constant scalar curvature on the blow-up according to [18]. We present a generalization of this construction to the case of parabolically polystable ruled surfaces. Thus we can produce numerous examples of Kähle...

In this article, a new type of ruled surfaces in a Lorentz 3-space R31 is obtained by a strictly connected time-like oriented line moving with Frenet s frame along a space-like curve. These surfaces are classified into time-like and space-like surfaces. The wellknown theorems due to Bonnet and Chasles in the 3-dimensional Euclidean space are proved for a time-like ruled surface. 2005 Published ...

We present an algorithm that covers any given rational ruled surface with two rational parametrizations. In addition, we present an algorithm that transforms any rational surface parametrization into a new rational surface parametrization without affine base points and such that the degree of the corresponding maps is preserved.