Abstract
There are three areas of study of geometric entities living in the three dimensional projective space over the complex numbers which have proved to be strictly connected: the classification of space curves, surfaces and vector bundles. Despite all the production in these areas, several key unsolved problems still challenge us. After the work of many classical and modern authors (see [H1], [H2] for a bibliography) the problem of classifying space curves has been divided into two main parts: 1) (the so called Halphen problem) Find all the triples of integers (n, g, s) such that there exists a smooth irreducible curve C ⊂ IP 3 of degree n and genus g lying on a surface of degree s but not on any surface of lower degree; 2) Study the properties of the Hilbert scheme Hn,g of smooth irreducible curves in IP , namely its irreducible components, dimension, singular points. Let G(n, s) be the maximum genus of a curve of degree n not contained in a surface of degree less than s. Then part 1 above falls in turn into three ranges for n and s:
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