On a class of rational cuspidal plane curves

We obtain new examples and the complete list of the rational cuspidal plane curvesC with at least three cusps, one of which has multiplicitydegC-2. It occurs that these curves are projectively rigid. We also discuss the general problem of projective rigidity of rational cuspidal plane curves.     

On limiting rational curves

This article was processed by the author using the Springer-Verlag T_EX mamath macro package 1990.     

Saturation of Jacobian ideals: Some applications to nearly free curves,line arrangements and rational cuspidal plane curves

In this note we describe the minimal resolution of the ideal $I_f$, the saturation of the Jacobian ideal of a nearly free plane curve $C:f=0$. In particular, it follows that this ideal $I_f$ can be generated by at most 4 polynomials. Related general results by Hassanzadeh and Simis on the saturation of codimension 2 ideals are discussed in detail. Some applications to rational cuspidal plane curves and to line arrangements are also given.     

On the number of cusps on cuspidal curves on Hirzebruch surfaces

In this article we give an upper bound for the number of cusps on a cuspidal curve on a Hirzebruch surface. We adapt the results that have been found for a similar question asked for cuspidal curves on the projective plane, and restate the results in this new setting.     

On the rational cuspidal subgroup and the rational torsion points of

For two distinct prime numbers , , we compute the rational cuspidal subgroup of and determine the -primary part of the rational torsion subgroup of the old subvariety of for most primes . Some results of Berkovic on the nontriviality of the Mordell-Weil group of some Eisenstein factors of are also refined.

     

On fair parametric rational cubic curves

First we derive conditions that a parametric rational cubic curve segment, with a parameter, interpolating to plane Hermite data {(x _i ^(k) ,y _i ^(k) ),i = 0, 1;k = 0, 1} contains neither inflection points nor singularities on its segment. Next we numerically determine the distribution of inflection points and singularities on a segment which gives conditions that aC ² parametric rational cubic curve interpolating to dataS = {(x _i ^(k) ,y _i ^(k) ), 0 i n} is free of inflection points and singularities. When the parametric rational cubic curve reduces to the well-known parametric cubic one, we obtain a theorem on the distribution of the inflection points and singularities on the cubic curve segment which has been widely used for finding aC ¹ fair parametric cubic curve interpolating toS.     

On special rational curves in Grassmannians

Numerical and geometric characterizations, among all morphisms , of those which are -equivalent to the canonical morphism induced by the Morita equivalence –, are presented. The author was partially supported by KBN grants 1P03A 036 26 and 115/E-343/SPB/6.PR UE/DIE 50/2005-2008. Received: 10 September 2005     

Remark on Tono's theorem about cuspidal curves

We give an upper bound for the number of cusps of a plane affine or projective curve via its first Betti number.     

Varieties of cuspidal curves in ℙ r

Supported by a CSIC Postdoctoral Fellowship     

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 variety of rational curves inP n

Summary In this note one defines a stratification of the variety of rational curves inP ⁿ of given degree in terms of the decomposition of the normal bundle (see [E-vdV], [G-S] for the casen=3). The strata are showed to be irreducible and their dimension is computed.