Related papers: Rational Cuspidal Curves
According to a conjecture of H. Clemens, the dimension of the space of rational curves on a general projective hypersurface should equal the number predicted by a na\"ive dimension count. In the case of a general hypersurface of degree 7 in…
By associating to a curve C of genus g=2k and a pencil of degree d=k+1 the so-called trace curve (resp. the reduced trace curve) we define a rational map from the Hurwitz space of admissible covers of genus g=2k and degree d=k+1 to a moduli…
This paper proposes a simple technique of curve and surface construction with B-splines. Given a control polygon or a control mesh together with node ordinates corresponding to all control points, a rational curve or surface is obtained by…
We study the rationality of the components of the conchoid to an irreducible algebraic affine plane curve, excluding the trivial cases of the isotropic lines, of the lines through the focus and the circle centered at the focus and radius…
By using analytic method, we prove that there exist rational curves on compact Hermitian manifolds with positive holomorphic bisectional curvature. It confirms a question of S.-T. Yau. It is well-known that Mori proved in \cite{Mori79} that…
The aim of these notes is to give a introduction to the ideas and techniques of handling rational curves on varieties. The main emphasis is on varieties with many rational curves which are the higher dimensional analogs of rational curves…
In this paper, we regard the smooth quadric threefold $Q_{3}$ as Lagrangian Grassmannian and search for fixed rational curves of low degree in $Q_{3}$ with respect to a torus action, which is the maximal subgroup of the special linear group…
We classify all complex surfaces with quotient singularities that do not contain any smooth rational curves, under the assumption that the canonical divisor of the surface is not pseudo-effective. As a corollary we show that if $X$ is a log…
First we solve the problem of finding minimal degree families on toric surfaces by reducing it to lattice geometry. Then we describe how to find minimal degree families on, more generally, rational complex projective surfaces.
We give a constructive proof for the existence of a unique rational motion of minimal degree in the dual quaternion model of Euclidean displacements with a given rational parametric curve as trajectory. The minimal motion degree equals the…
This note (which makes no claim to novelty) presents a proof of the separable rational connectedness of smooth cubic hypersurfaces, in any characteristic, by showing how to explicitly construct very free curves (of degree 3) on them. -----…
We study the existence of some irreducible projective plane curves of degree~$8$ with some prescribed topological type of singularities in the algebraic and symplectic worlds.
It is proved that a smooth rational surface in projective four-space, which is ruled by cubics or quartics has degree at most 12. It is also proved that a smooth rational surface in projective four-space which is the image of Fn by a linear…
A smooth Hermitian surface $X$ is a projective surface isomorphic to the Fermat surface of degree $q+1$ in positive characteristic. We study incidence relations of the rational curves of degree $q+1$ contained in $X$, and show that such…
This paper is concerned with rational curves on real classical groups. Our contributions are three-fold: (i) We determine the structure of quadratic rational curves on real classical groups. As a consequence, we completely classify…
This article, based on the talk given by one of the authors at the Pierrettefest in Castro Urdiales in June 2008, is an overview of a number of recent results on the polar invariants of plane curve singularities.
This paper reviews known results on the rigid isotopy classification of plane curves of degree $m\leq6$ and curves of small degrees on quadrics. The paper's study completes the rigid isotopy classification of nonsingular real algebraic…
This article is a comprised version of my doctoral thesis defended at Basel University in 1995. It concerns circle packings on compact surfaces of any genus.
These are notes on some algebraic geometry of complex projective curves, together with an application to studying the contact curves in CP^3 and the null curves in the complex quadric Q^3 in CP^4, related by the well-known Klein…
The ruled surfaces, i.e., surfaces generated by one parametric set of lines, are widely used in the~field of applied geometry. An~isophote on a surface is a curve consisting of surface points whose normals form a constant angle with some…