BOUNDING FAMILIES OF RULED SURFACES

In this paper we provide a sharp bound for the dimension of a family o f ruled surfaces of degree d in P-K(3). We also find the families with maximal dimension: the family of ruled surfaces containing two unisec ant skew lines, when d greater than or equal to 9 and the family of ra tional ruled surfaces, when d less than or equal to 9. The first tool we use is a Castelnuovo-type bound for the irregularity of ruled surfa ces in P-K(n). The second tool is an exact sequence involving the norm al sheaf of a curve in the grassmannian. This sequence is analogous to the one constructed by Eisenbud and Harris in 1992, where they deal w ith the problem of bounding families of curves in projective Space. Ho wever, our construction is more general since we obtain the mentioned sequence by purely algebraic means, studying the geometry of ruled sur faces and of the grassmannian.