Geometric and cohomological properties of rank 2 vector bundles on curves

Here we study the postulation of curves embedded in a smooth quadric hypersurface of P ⁴ and P ⁵ and relate this subject to the study of cohomological properties of rank 2 spanned vector bundles on smooth projective curves. Received: May 25, 2000; in final form: January 3, 2001?Published online: May 29, 2002     

On the variety of rational space curves

We consider the locus of smooth rational curves of given degree in a given projective space, which are incident to a generic collection of linear spaces. When this locus is finite (resp. 1-dimensional) we give a recursive procedure to compute its degree (resp. geometric genus). The method is based on the elementary geometry of ruled surfaces.     

On the varieties parametrizing rational space curves with fixed normal bundle

On the one hand this note complements the results obtained in [2] for rational curves, and, on the other hand, it introduces the study of the normal bundle of a curve of arbitrary genus in IP³ (the projective space over an algebraically closed field of characteristic 0).The most important result concerns rational curve; we give the dimension of the variety parametrizing rational curve of fixed degree and fixed normal bundle, and we show they are quasi-projective, integral and Cohen-Macaulay.This work was carried out while the author held a CNR research fellowship at the Institute of Mathematics, University of Oslo     

On certain cohomological invariants of groups

Let R be a left and right ℵ₀-Noetherian ring. We show that if all projective left and all projective right R-modules have finite injective dimension, then all injective left and all injective right R-modules have finite projective dimension. Using this result, we prove that the invariants and , which were introduced by Gedrich and Gruenberg (1987) [15], are equal for any group G. As an application of the latter equality, we show that a group G is finite if and only if , where is the generalized cohomological dimension of groups introduced by Ikenaga (1984) [21].     

On the Halphen transform of algebraic space curves

The Halphen transform of a plane curve is the curve obtained by intersecting the tangent lines of the curve with the corresponding polar lines with respect to some conic. This transform was introduced by Halphen as a branch desingularization method in [5 Halphen, G. H. (1876). Sur une série de courbes analogues aux développées. J. Math. Pures Appl. 3e série, tome 2:87–144. [Google Scholar]] and has also been studied in [2 Coolidge, J. L. (2004). A Treatise on Algebraic Plane Curves. New York: Dover Publications, Inc., 1959, xxiv+513 pp. [Google Scholar], 8 Josse, A. (1995). Transformation d’Halphen. (French) [The Halphen transform]. Commun. Algebra 23(12):4343–4364.[Taylor & Francis Online] , [Google Scholar]]. We extend this notion to the Halphen transform of a space curve and study several of its properties (birationality, degree, rank, class, desingularization).     

On curves with a theta-characteristic whose space of sections has dimension 4

Partially supported by the 40% Project ALGEBRAIC GEOMETRY and by the CNR     

Pythagorean-hodograph space curves

We investigate the properties of polynomial space curvesr(t)={x(t), y(t), z(t)} whose hodographs (derivatives) satisfy the Pythagorean conditionx′²(t)+y′²(t)+z′²(t)≡σ²(t) for some real polynomial σ(t). The algebraic structure of thecomplete set of regular Pythagorean-hodograph curves in ℝ³ is inherently more complicated than that of the corresponding set in ℝ². We derive a characterization for allcubic Pythagoreanhodograph space curves, in terms of constraints on the Bézier control polygon, and show that such curves correspond geometrically to a family of non-circular helices. Pythagorean-hodograph space curves of higher degree exhibit greater shape flexibility (the quintics, for example, satisfy the general first-order Hermite interpolation problem in ℝ³), but they have no “simple” all-encompassing characterization. We focus on asubset of these higher-order curves that admits a straightforward constructive representation. As distinct from polynomial space curves in general, Pythagorean-hodograph space curves have the following attractive attributes: (i) the arc length of any segment can be determined exactly without numerical quadrature; and (ii) thecanal surfaces based on such curves as spines have precise rational parameterizations.     

On the space curves with the same image under the gauss maps

From an irreducible complete immersed curveX in a projective space ? other than a line, one obtains a curveX ^′ in a Graasmann manifoldG of lines in ? that is the image ofX under the Gauss map, which is defined by the embedded tangents ofX. The main result of this article clarifies in case of positive characteristic what curvesX have the sameX′: It is shown thatX is uniquely determined byX′ ifX, or equivalentlyX′, has geometric genus at least two, and that for curvesX ₁ andX ₂ withX ₁ ≠X ₂ in ?, ifX′₁ =X′₂ inG and eitherX ₁ orX ₂ is reflexive, then bothX ₁ andX ₂ are rational or supersingular elliptic; moreover, examples of smoothX ₁ andX ₂ in that case are given.     

General properties of Bertrand curves in Riemann–Otsuki space

In this paper we studied the general properties of Bertrand curves and their characterizations in Riemann–Otsuki space.