An explicit construction of ruled surfaces

Abstract

a b s t r a c tThe main goal of this paper is to give a general algorithm to compute, via computer-algebra systems, an explicit set of generators of the ideals of the projective embeddings of ruled surfaces, i.e. projectivizations of rank two vector bundles over curves, such that the fibers are embedded as smooth rational curves.There are two different applications of our algorithm. Firstly, given a very ample linear system on an abstract ruled surface, our algorithm allows computing the ideal of the embedded surface, all the syzygies, and all the algebraic invariants which are computable from its ideal as, for instance, the k-regularity. Secondly, it is possible to prove the existence of new embeddings of ruled surfaces.The method can be implemented over any computer-algebra system able to deal with commutative algebra and Gröbner-basis computations. An implementation of our algorithms for the computer-algebra system Macaulay2 (cf. [Daniel R. Grayson, Michael E. Stillman, Macaulay 2, a software system for research in algebraic geometry, 1993. Available at http://www.math.uiuc.edu/Macaulay2/]) and explicit examples are enclosed.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 0 (C, E) = 0 but H 0 (C, E ⊗ G) = 0 for any line bundle G of negative degree, then E fits into a short exact sequenceWe consider the geometrically ruled surface X := P(E), endowed with the natural projection p : P(E) → C . In this case Pic(X ) ∼ = Z ⊕ p * Pic(C ), where Z is generated by the tautological divisor of X , i.e. a divisor C 0 , image of a section σ 0 : C → X with minimal self-intersection. According to this notation, every divisor on X is linearly (resp. numerically) equivalent toWe choose a very ample divisor A on X and we consider the polarized ruled surface (X, A), i.e., X embedded in P h 0 (X,A)−1 by |A|: we aim to give an algorithm to compute a set of generators of its ideal I X in the ring S(V ) := ⊕ i≥0 S i (V ), the symmetric algebra of V = H 0 (X, A). The algorithm requires the knowledge of the following data: a set of generators of the ideal I C of any embedded image of C in some projective space; the (Weil) divisor B of C ; and the extension class giving $ This work was written within the framework of the national research project ''Geometry of Algebraic Varieties'' Cofin 2006 of MIUR and the DFGForschungsschwerpunkt ''Globale Methoden in der komplexen Geometrie''.