Classification of Incidence Scrolls (II)

The aim of this paper is to obtain a classification of the scrolls in Pn which are defined by a one-dimensional family of lines meeting a certain set of linear spaces in Pn, a first classification for genus 0 and 1 is given in paper [1]. These ruled surfaces are called incidence scrolls, and such an indicated set is a base of the incidence scroll. In particular, we compute its degree and genus. For this, we define the fundamental incidence scroll to be the scroll in Pn formed by the lines which meet (2n-3) P{n-2}'s in general position. Then all the others incidence scrolls will be portions of degenerate forms of this. In this way, we can obtain all the incidence scrolls in Pn, n>2, with base in general position.Comment: 22 page

Classification of Incidence Scrolls(I)

The aim of this paper is to obtain a classification of scrolls of genus 0 and 1, which are defined by a one-dimensional family of lines meeting a certain set of linear spaces in ${\bf P}^n$. These ruled surfaces will be called incidence scrolls and such a set will be the base of the incidence scroll. Unless otherwise stated, we assume that the base spaces are in general position.Comment: Latex209, 19 pages, revised and corrected version. To appear in Manuscripta Mathematic

Another classification of Incidence Scrolls

The aim of this paper is the computation of the degree and genus of all incidence scrolls in Pn. For this, we fix the dimension of a linear space which have a base space of this fixed dimension. In this way, we can obtain all the incidence scrolls with a line as directrix curve, those whose base contains a plane, and so on.Comment: Latex, 10 pages. It contains three tables. Accepted for publication in Archiv der Mathemati

The Projective Theory of Ruled Surfaces

The aim of this paper is to get some results about ruled surfaces which configure a projective theory of scrolls and ruled surfaces. Our ideas follow the viewpoint of Corrado Segre, but we employ the contemporaneous language of locally free sheaves. The results complete the exposition given by R. Hartshorne and they have not appeared before in the contemporaneous literature.Comment: 40 page

The generic special scroll of genus g in Pn. Special scrolls in P3

We study the generic linearly normal special scroll of genus g in P^N. Moreover, we give a complete classification of the linearly normal scrolls in P^3 of genus 2 and 3.Comment: Submitted for publicatio

Involutions of a canonical curve

We give a geometrical characterization of the ideal of quadrics containing a canonical curve with an involution. This implies to study involutions of rational normal scrolls and Veronese surfaces.Comment: 20 page

Dimension of the Moduli Space of curves with an involution

Given a smooth curve X of genus g we compute de dimension of the family of curves C which have an involution over X. Moreover we distinguish when the curve C is hyperelliptic.Comment: 7 page

Projective normality of special scrolls II

We study the projective normality of a linearly normal special scroll R of degree d and speciality i over a smooth curve X of genus g. We relate it with the Clifford index of the base curve X. If d>=4g-2i-Cliff(X)+1, i>=3 and R is smooth, we prove that the projective normality of the scroll is equivalent to the projective normality of its directrix curve of minimum degree.Comment: 7 page

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 Teorem: any special scroll is the projection of a canonical scroll and they allow to understand the classification of special scrolls in P3. 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.Comment: Latex2.09; 32 page

Projective generation and smoothness of congruences of order 1

In this paper we give the projective generation of congruences of order 1 of r-dimensional projective spaces in P^N from their focal loci. In a natural way, this construction shows that the corresponding surfaces in the grassmannian are the Veronese surface, and rational ruled surfaces eventually with singularities. We characterize when these surfaces are smooth, recovering and generalizing a Ziv Ran's result.Comment: Latex2e, 18 page