Bounding families of ruled surfaces

In this paper we provide a sharp bound for the dimension of a family of ruled surfaces of degree $d$ in $\mathbf P_{\mathbf K}^3$. We also find the families with maximal dimension: the family of ruled surfaces containing two unisecant skew lines, when $d\ge 9$ and the family of rational ruled surfaces, when $d\le 9$.

The first tool we use is a Castelnuovo-type bound for the irregularity of ruled surfaces in $\mathbf P_{\mathbf K}^n$. The second tool is an exact sequence involving the normal sheaf of a curve in the grassmannian. This sequence is analogous to the one constructed by Eisenbud and Harris in 1992, where they deal with the problem of bounding families of curves in projective space. However, our construction is more general since we obtain the mentioned sequence by purely algebraic means, studying the geometry of ruled surfaces and of the grassmannian.