are surjective for all m ≥ 2. If L is normally generated, then we say that L satisfies property Np, if the matrices in the free resolution of R over S have linear entries until the pth stage. In particular, property N1 says that the homogeneous ideal I of X in P(H(L)) is generated by quadrics. A line bundle satisfying property N1 is also called normally presented. Let R = k⊕R1 ⊕R2 ⊕ . . . be a ...

Let F be a differentiable manifold endowed with an almost Kähler structure (J, ω), α a J-holomorphic action of a compact Lie groupˆK on F , and K a closed normal subgroup ofˆK which leaves ω invariant. The purpose of this article is to introduce gauge theoretical invariants for such triples (F, α, K). The invariants are associated with moduli spaces of solutions of a certain vortex type equatio...

This paper presents a positioning strategy for flank milling ruled surfaces. It is a modification of a positioning method developed by Bedi et al. [1]. A cylindrical cutting tool is initially positioned tangential to the two boundary curves on a ruled surface. Optimization is used to move these tangential points to different curves on the ruled surface to reduce the error. A second optimization...

Chen, Sederberg, and Zheng introduced the notion of a μ-basis for a rational ruled surface in Chen et al. (2001) and showed that its resultant is the implicit equation of the surface, if the parametrization is generically injective. We generalize this result to the case of an arbitrary parametrization of a rational ruled surface. We also give a new proof for the corresponding theorem in the cur...

We address the following question. Consider a sub-manifold of an affine space, defined by its equations F = 0: does there exist a finite characterization of ruled sub-manifolds in terms of derivatives of F via algebraic inequalities and/or equalities ? This question is related to an open problem in control theory: the finite characterization of flat systems [2] and, more generally, of systems l...

The purpose of this article is to study the minimal free resolution of homogeneous coordinate rings of elliptic ruled surfaces. Let X be an irreducible projective variety and L a very ample line bundle on X , whose complete linear series defines the morphism φL : X −→ P(H (L)) Let S = ⊕∞ m=0 S H(X,L) and R(L) ⊕∞ m=0 H (X,L). Since R(L) is a finitely generated graded module over S, it has a mini...

The main goal of this paper is to give a general method to compute (via computer algebra systems) an explicit set of generators of the ideals of the projective embeddings of some ruled surfaces, namely projective line bundles over curves such that the fibres are embedded as smooth rational curves. Indeed, although the existence of the embeddings that we consider is well known, often in literatu...

New historical aspects of the classification, by Cayley and Cremona, of ruled quartic surfaces and the relation to string models and plaster models are presented. In a ‘modern’ treatment of the classification of ruled quartic surfaces the classical one is corrected and completed. A conceptual proof is presented of a result of Rohn concerning curves in P × P of bi-degree (2, 2). The string model...

Problems linking the geometry of two-dimensional manifolds and the spectrum of associated Laplacians have been considered for more than a century. While classical motivations come from theories of elasticity and electromagnetism, the same rather simple models can be also remarkably successful in describing even rather complicated phenomena in quantum heterostructures. Here, an enormous amount o...