Analytic sphere eversion using ruled surfaces

Sphere eversions have been described so far by either pictures with minimal topological complexity, numerical evolution or complex equations. We write down relatively simple explicit formulas for the whole eversion, both analytic and topologically simpler, including also Boy surface (real projective plane), using a family of ruled surfaces. We show their usefulness in visualizing the process using commonly available modeling software.     

Ruled minimal surfaces in the Berger sphere

We show that any ruled minimal surface in the Berger sphere is a helicoid whose axis is a Hopf fiber by solving the ruled minimal surface equation in the parametric form.     

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 Projective Normality of Smooth Surfaces of Degree Nine

The projective normality of smooth, linearly normal surfaces of degree 9 in ^N is studied. All nonprojectively normal surfaces which are not scrolls over a curve are classified. Results on the projective normality of surface scrolls are also given.     

Special curves on ruled surfaces in Galilean and pseudo-Galilean spaces

We develop the theory of ruled surfaces in the pseudo-Galilean space G ₃ ¹. Furthermore, we prove the Bonnet theorem and investigate the osculating quadrics for ruled surfaces in G ₃ ¹as well as in the Galilean space G ₃. This revised version was published online in June 2006 with corrections to the Cover Date.     

Projective Normality of Special Scrolls

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.     

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.

     

On Rational Surfaces Ruled by Conics

Abstract

We study projective rational surfaces ruled by conics, describing their singularities and special fibres. In particular, if Sis smooth, we give a “canonical” procedure to determine a minimal model among the geometrically ruled surfaces birational to S.     

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.     

Surfaces in $${\mathbb {P}^{4}}$$ fibered in cubics

Any algebraic surface in which is fibered in cubics, so that the generic fibre is a twisted cubic, gives rise to a curve Γ in a suitable compactification X of the space of smooth rational cubics of In this paper the case n = 4 is addressed and the corresponding space X is studied. We apply our results to complete the classification of smooth, rational surfaces in ruled in cubics. This work is within the framework of the national research project “Geometry on Algebraic Varieties” Cofin 2006 of MIUR.     

Canonical measures on the moduli spaces of compact Riemann surfaces

We study some explicit relations between the canonical line bundle and the Hodge bundle over moduli spaces for low genus. This leads to a natural measure on the moduli space of every genus which is related to the Siegel symplectic metric on Siegel upper half-space as well as to the Hodge metric on the Hodge bundle.     

Canonical vector heights on K3 surfaces with Picard number three-- An argument for nonexistence

In this paper, we investigate a K3 surface with Picard number three and present evidence that strongly suggests a canonical vector height cannot exist on this surface.

     

Zero cycles on certain surfaces in arbitrary characteristic

Letk be a field of arbitrary characteristic. LetS be a singular surface defined overk with multiple rational curve singularities and suppose that the Chow group of zero cycles of its normalisation is finite dimensional. We give numerical conditions under which the Chow group of zero cycles ofS is finite dimensional.     

Laguerre homogeneous surfaces in R~3

In this paper,we study oriented surfaces of R3 in the context of Laguerre geometry.We construct Laguerre invariants on the non-Dupin developable surfaces,which determine the surfaces up to a Laguerre transformation.Finally,we classify the Laguerre homogeneous surfaces in R3 under the Laguerre transformation groups.     

Intersection theory over quantum ruled surfaces

In this paper, we will extend several results on intersection theory over commutative ruled surfaces to quantum ruled surfaces. Typically, we define the fiber of a closed point, the quasi-section, and the quasi-canonical divisor on a quantum rules surface, and study how these “curves” on a quantum ruled surface intersect with each other.     

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.