In this survey, we report on progress concerning families of projective curves with fixed number and fixed (topological or analytic) types of singularities. We are, in particular, interested in numerical, universal and asymptotically proper sufficient conditions to guarantee the nonemptyness, T-smoothness and irreducibility of the variety of all projective curves with prescribed singularities in a fixed linear system. We also discuss the analogous problem for hypersurfaces of arbitrary dimension with isolated singularities, and we close with a section on open problems and conjectures.

Equisingular Families of Projective Curves

We describe degenerations of projective plane curves to curves containing a fixed line $l$ as a component, and show that $H^1({\\overline V}_{n,d,m}, {\\Cal O} (r))=0, r \\in{\\Bbb Z}$, where $V_{n,d,m}\\subset {\\Bbb P}^N (N = n(n+3)/2)$ is the subscheme consisting of irreducible plane curves having smooth contact of order at least m with $l$ at a fixed point p of $l$ and d nodes and no other singularities. Note: the notion of admissible schemes of plane curves, introduced for the proof of the vanishing theorem, allows us to give a recipe for calculating the Hilbert polynomial of ${\\overline V}_{n,d}$ (see Sect. 4), in particular the quantum cohomology of the plane.

/pdf/local-properties-of-families-of-plane-curves-lly9pisf96.pdf

Local properties of families of plane curves

We describe degenerations of projective plane curves to curves containing a fixed line $l$ as a component, and show that $H^1({\overline V}_{n,d,m}, {\Cal O} (r))=0, r \in{\Bbb Z}$, where $V_{n,d,m}\subset {\Bbb P}^N (N = n(n+3)/2)$ is the subscheme consisting of irreducible plane curves having smooth contact of order at least m with $l$ at a fixed point p of $l$ and d nodes and no other singularities. Note: the notion of admissible schemes of plane curves, introduced for the proof of the vanishing theorem, allows us to give a recipe for calculating the Hilbert polynomial of ${\overline V}_{n,d}$ (see Sect. 4), in particular the quantum cohomology of the plane.

Global properties of families of plane curves

If an algebraic curve C (irreducible and reduced) moving in a family in projective n-space specializes into a curve C0, having associated cycle Z=m1B1+···+mrBr, then the geometric genera g, g1,....,gr of C, B1,...,Br respectively and the coefficients m1,...,mr must satisfy a certain inequality (found by Albanese). The realization (or existence) problem asks whether an inequality of this type actually arises from an algebraic family of curves. In this paper some results are obtained concerning the strong version of the problem, where one specifies the cycle Z; with these an affirmative solution of the weak version (where the components Bi are not specified) is obtained.

On the geometric realization of Albanese's inequality

First, a modern presentation of the theory of the Halphen transform is given. This method associates to a plane projective curve C, once a general conic has been chosen, another birationally equivalent plane curve, whose singularities are simpler than those of C. Repeating, a curve is obtained whose only singularities are nodes. Next, it is studied how to apply this process to a family of plane curves. With this technique it is possible to transform a given family (with irreducible general member) into one where, generically, the curves are nodal. Finally, it is studied a similar process, called the Halphen–Picard transformation, for surfaces in three-space. By suitably reiterating this procedure, a surface can be transformed into a birationally equivalent one (in the same projective space), such that the sections with planes in a general pencil are, generically, nodal curves.

On Some Classical Methods to Improve Algebraic Curves and Surfaces

© Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1987, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Genera of curves varying in a family

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Genera of curves varying in a family