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.     

Algebraic non ruled surfaces with sectional genus equal to seven

Summary Let (X, L) be a pair consisting of a smooth, algebraic, non-ruled surfaceX and a very ample line bundleL on it. In this paper we classifyX in the casesď=2g−i, wherei=2, … 5,g is the sectional genus ofX andď is the degree of the minimal reduction ofX. We also give good estimates of the irregularityq and the geometric genusp _g. In particular we apply our results to the caseg=7.

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Riassunto Sia (X, L) la coppia formata da una superficie algebrica, liscia, non-rigataX e un fibrato di retteL molto ampio suX. In questo lavoro diamo una classificazione diX nel caso in cuiď=2g−i, dovei=2, … 5,q è il genere sezionale diX eď è il grado della riduzione minimale diX. Diamo anche delle stime dell'irregolaritàq e del genere geometricop _g diX. In particolare applichiamo i nostri risultati al casoq=7.

     

Vector bundles on ruled Fano 3-folds

Let be a ruled Fano 3-fold. The goal of this paper is to compute the dimension, prove the irreducibility and smoothness and describe the structure of the moduli space M _L (2;c ₁,c ₂) of L-stable, rank 2 vector bundles E on X with certain Chern classes and for a suitable polarization L closely related to c ₂. More precisely, we will cover the study of some moduli spaces M _L (2;c ₁,c ₂) such that the generic point is given as a non-trivial extension of line bundles. This work nicely reflects the general philosophy that moduli spaces inherits a lot of geometrical properties of the underlying variety. Received: 16 February 1999 / Revised version: 2 July 1999     

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.