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 (cf. [G-S]) are included, so that anyone can use them for his own purposes. Introduction and Notation Let E be a rank 2 vector bundle over a smooth, genus q, curve C. It is known that any such vector bundle E, regarded as a sheaf, is an extension of invertible sheaves. If E is a normalized vector bundle, i.e. H(C,E) 6= 0 but H(C,E ⊗ G) = 0 for any line bundle G of negative degree, then E sits into a short exact sequence (0.1) 0 → OC → E → L → 0, and L = det(E). We now consider the geometrically ruled surface X := P(E), endowed with the natural projection p : P(E) → C. In this case Pic(X) ∼= Z ⊕ pPic(C), where Z is generated by the tautological divisor of X , i.e. a divisor C0, image of a section σ0 : C → X , whose associated invertible sheaf is OX(1). According to this notation, every divisor on X is linearly (resp. numerically) equivalent to aC0 + p B (resp. aC0 + bf , being f a fiber of p) where B is a degree b divisor of C. We choose a very ample divisor A on X and we consider the polarized ruled surface (X,A): what are then the equations ofX? In other words,X is embedded in P (X,A)−1 by |A| and we aim to give an algorithm for computing a set of generators of the ideal IX in the ring S(V ) := ⊕i≥0S (V ), the symmetric algebra of V = H(X,A). Ampleness conditions for the divisor A are classical and well known (cf. e.g. [H]). In particular, by the Nakai’s criterion, denoting with e := − degE the invariant of X , an ample divisor A is numerically equivalent to aC0 + bf with a ≥ 1 and b > ae if e ≥ 0 or b > ae 2 if e < 0, and the very ampleness for A should be checked case by case with some criteria, e.g. Reider’s criterion (cf. [R]) or by looking at the image of X by |A|. 1991 Mathematics Subject Classification. 14J26, 14Q10. This work is within the framework of the national research project “Geomety of Algebraic Varieties” Cofin 2004 of MIUR and the DFG-Forschungsschwerpunkt “Globale Methoden in der komplexen Geometrie”.