An explicit construction of ruled surfaces

The main goal of this paper is to give a general method to compute (via computer algebra systems) an explicit set of generators of the ideals of the projective embeddings of some ruled surfaces, namely projective line bundles over curves such that the fibres are embedded as smooth rational curves. Indeed, although the existence of the embeddings that we consider is well known, often in literature there are no explicit descriptions of the corresponding projective ideals. Such an explicit description allows to compute, besides all the syzygies, some of the important algebraic invariants of the surface, for instance the $k$-regularity, which are not always easy to compute by general formulae or by geometric arguments. An implementation of our algorithms and explicit examples for the computer algebra system Macaulay2 are included, so that anyone can use them for his own purposes.