On closed timelike and spacelike ruled surfaces

In this study, by using the concepts and results on spherical curves in dual Lorentzian space, we give the criterions for ruled surfaces with non‐lightlike ruling to be closed (periodic). Moreover, we obtain the necessary and sufficient conditions to guarantee that a timelike or a spacelike ruled surface is closed (periodic). Copyright © 2016 John Wiley & Sons, Ltd.     

On the integral invariants of a closed ruled surface

Summary In this paper, it is shown that the dual integral invariant of a closed ruled surface, the dual angle of pitch, corresponds to the dual spherical surface area described by the dual spherical indicatrix of the closed ruled surface. So, new geometric interpretations of the real angle of pitch and the real pitch of a closed ruled surface, and some results are given.     

The degree of the variety of rational ruled surfaces and Gromov-Witten invariants

We compute the degree of the variety parametrizing rational ruled surfaces of degree in by relating the problem to Gromov-Witten invariants and Quantum cohomology.

     

Triple-point defective ruled surfaces

In [L. Chiantini, T. Markwig, Triple-point defective regular surfaces. arXiv:0705.3912, 2007] we studied triple-point defective very ample linear systems on regular surfaces, and we showed that they can only exist if the surface is ruled. In the present paper we show that we can drop the regularity assumption, and we classify the triple-point defective very ample linear systems on ruled surfaces.     

Canonical geometrically ruled surfaces

We prove the existence of canonical scrolls; that is, scrolls playing the role of canonical curves. First of all, they provide the geometrical version of Riemann Roch Theorem: any special scroll is the projection of a canonical scroll and they allow to understand the classification of special scrolls in P^N. Canonical scrolls correspond to the projective model of canonical geometrically ruled surfaces over a smooth curve. We also prove that the generic canonical scroll is projectively normal except in the hyperelliptic case and for very particular cases in the nonhyperelliptic situation. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Approximation by ruled surfaces

Given a surface or scattered data points from a surface in 3-space, we show how to approximate the given data by a ruled surface in tensor product B-spline representation. This leads us to a general framework for approximation in line space by local mappings from the Klein quadric to Euclidean 4-space. The presented algorithm for approximation by ruled surfaces possesses applications in architectural design, reverse engineering, wire electric discharge machining and NC milling.     

General blowups of ruled surfaces

In this note we study blowups of algebraic surfaces of Kodaira dimension κ = - ∞ at general points, their embeddings and secant varieties of the embedded surfaces.     

Anticanonical models of ruled surfaces

Summary One investigates the properties of the so-called anticanonical model of a ruled non-rational surfaceX with the anticanonical dimension 2 (see theorem 2 below). In such a way one gets a ?non-rational? analogue of a result of F. Sakai (see [12], and also theorem 1 below). As an application of these two theorems one characterizes the normal projective surfacesY overC with the property that there is a positive integerr such that—rK _Y is an ample Cartier divisor, whereK _y is a canonical (Weil) divisor ofY (see corollary 8 below).

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Riassunto Si studiano i modelli anticanonici delle superficie rigate non-razionali di dimensione anticanonica 2 (teorema 2). Si trova un risultato simile a quello di F. Sakai (teorema 1). Come applicazione si dà una caratterizzazione delle superficie proiettive normaliY sul campoC tali che un multiplo—rK _Y, conr>0, è un divisore di Cartier ampio, doveK _y è un divisore (di Weil) canonico (Corollario 8).

     

Bounding families of ruled surfaces

In this paper we provide a sharp bound for the dimension of a family of ruled surfaces of degree in . We also find the families with maximal dimension: the family of ruled surfaces containing two unisecant skew lines, when and the family of rational ruled surfaces, when .

The first tool we use is a Castelnuovo-type bound for the irregularity of ruled surfaces in . The second tool is an exact sequence involving the normal sheaf of a curve in the grassmannian. This sequence is analogous to the one constructed by Eisenbud and Harris in 1992, where they deal with the problem of bounding families of curves in projective space. However, our construction is more general since we obtain the mentioned sequence by purely algebraic means, studying the geometry of ruled surfaces and of the grassmannian.

     

Projective normality of ruled surfaces

In this article we study normal generation of irrational ruled surfaces. When is a smooth curve of genus , Green and Lazarsfeld proved that a very ample line bundle    Pic with deg   Cliff is normally generated where Cliff denotes the Clifford index of the curve (Green and Lazarsfeld, 1986). We generalize this to line bundles on a ruled surface over .

     

Ulrich bundles on ruled surfaces

In this short note, we study the existence problem for Ulrich bundles on polarized ruled surfaces, focusing our attention on the smallest possible rank. We show that existence of Ulrich line bundles occurs if and only if the coefficient α of the minimal section in the numerical class of the polarization equals one. For other polarizations, we prove the existence of rank two Ulrich bundles.     

Anti de Sitter horospherical flat timelike surfaces

We investigate a special timelike surfaces in Anti de Sitter 3-space.We call such a timelike surface an Anti de Sitter horospherical flat surface which belongs to a class of surfaces given by one parameter families of Anti de Sitter horocycle.We give a generic classification of singularities and study the geometric properties of such surfaces from the viewpoint of Legendrian singularity theory.     

An explicit construction of ruled surfaces

The main goal of this paper is to give a general algorithm to compute, via computer-algebra systems, an explicit set of generators of the ideals of the projective embeddings of ruled surfaces, i.e. projectivizations of rank two vector bundles over curves, such that the fibers are embedded as smooth rational curves.There are two different applications of our algorithm. Firstly, given a very ample linear system on an abstract ruled surface, our algorithm allows computing the ideal of the embedded surface, all the syzygies, and all the algebraic invariants which are computable from its ideal as, for instance, the k-regularity. Secondly, it is possible to prove the existence of new embeddings of ruled surfaces.The method can be implemented over any computer-algebra system able to deal with commutative algebra and Gröbner-basis computations. An implementation of our algorithms for the computer-algebra system Macaulay2 (cf. [Daniel R. Grayson, Michael E. Stillman, Macaulay 2, a software system for research in algebraic geometry, 1993. Available at http://www.math.uiuc.edu/Macaulay2/]) and explicit examples are enclosed.     

On higher syzygies of ruled surfaces

We study higher syzygies of a ruled surface over a curve of genus with the numerical invariant . Let    Pic be a line bundle in the numerical class of . We prove that for , satisfies property if and , and for , satisfies property if and . By using these facts, we obtain Mukai-type results. For ample line bundles , we show that satisfies property when and or when and . Therefore we prove Mukai's conjecture for ruled surface with . We also prove that when is an elliptic ruled surface with , satisfies property if and only if and .