Classification of Elliptic Line Scrolls
نویسنده
چکیده
Introduction: Through this paper, a geometrically ruled surface, or simply a ruled surface, will be a P-bundle over a smooth curve X of genus g. It will be denoted by π : S = P(E0)−→X and we will follow the notation and terminology of R. Hartshorne’s book [3], V, Section 2. We will suppose that E0 is a normalized sheaf and X0 is the section of minimum self-intersection that corresponds to the surjection E0−→OX(e)−→0, ∧2 E ∼= OX(e). Consider the following question: which are the linear equivalence classes D ∼ mX0 + bf , b ∈ Pic(X), that correspond to very ample divisors? When g = 1 and m = 1, a characterization is known ([3], V, Ex. 2.12), but the classification of elliptic scrolls obtained by Corrado Segre in [5] does not follow directly from this. A scroll is the image of a ruled surface π : S = P(E0)−→X by a unisecant complete linear system.
منابع مشابه
A Remark on the Geometry of Elliptic Scrolls and Bielliptic Surfaces
The union of two quintic elliptic scrolls in P intersecting transversally along an elliptic normal quintic curve is a singular surface Z which behaves numerically like a bielliptic surface. In the appendix to the paper [ADHPR] where the equations of this singular surface were computed, we proved that Z defines a smooth point in the appropriate Hilbert scheme and that Z cannot be smoothed in P. ...
متن کاملON THE k-REGULARITY OF SOME PROJECTIVE MANIFOLDS
The conjecture on the (degree − codimension+ 1)regularity of projective varieties is proved for smooth linearly normal polarized varieties (X, L) with L very ample, for low values of ∆(X, L) = degree− codimension− 1. Results concerning the projective normality of some classes of special varieties including scrolls over curves of genus 2 and quadric fibrations over elliptic curves, are proved.
متن کاملOn Families of Rank-2 Uniform Bundles on Hirzebruch Surfaces and Hilbert Schemes of Their Scrolls
Several families of rank-two vector bundles on Hirzebruch surfaces are shown to consist of all very ample, uniform bundles. Under suitable numerical assumptions, the projectivization of these bundles, embedded by their tautological line bundles as linear scrolls, are shown to correspond to smooth points of components of their Hilbert scheme, the latter having the expected dimension. If e = 0, 1...
متن کاملDynamic Meshing Characteristics of Elliptic Cylinder Gear Based on Tooth Contact Analysis
As the most important working area of gear, teeth play the role of transmitting load and power. Tooth line and tooth profile are the two main characteristics of the tooth surface, which affect the shape of the tooth surface, tooth meshing characteristics and contact characteristics. Taking the elliptical cylinder gear pair in the reversing device of a new type of drum pumping unit as the resear...
متن کاملOn the Projective Normality of Smooth Surfaces of Degree Nine
The projective normality of smooth, linearly normal surfaces of degree 9 in P is studied. All non projectively normal surfaces which are not scrolls over a curve are classified. Results on the projective normality of surface scrolls are also given.
متن کامل