Applications of Some Properties of the Canonical Module in Computational Projective Algebraic Geometry

Introduction. The aim of this article is to show how one may use interesting properties of the canonical module for computational applications. It seems that the ideas we develop here are not used in computational algebraic geometry today. We do not claim any major improvement on the theoretical complexity of the problems we address, however we have the feeling that this alternative approach for projective schemes is quite promising, for two reasons. Firsts the algorithms are extremely simple, and everything may be computed either via Gröbner basis computations or linear algebra in the polynomial ring (which may have some advantages, as pointed out by J.C. Faugère). Secondly, the complexity of the algorithms are essentially controlled by the complexity of the output : for example the Castelnuovo-Mumford regularity of the top dimensional part and of its dualizing module, in the computation of the top dimensional component. We give here some bounds for the Castelnuovo-Mumford regularity in small dimension; much more is done in [6] giving greater evidence for “reasonable” bounds on the regularity of the top dimensional component, and thereby on the complexity of our algorithms. In a first part we recall, sometimes with (sketch of) proofs, the classical results that we will need in the sequel. We then give in part 2 a first example of applications by showing how a result of Hochster (and local duality) gives an easy way to compute the dimension of a projective scheme from generators of the defining ideal. This is a classical problem, see for example [11] or [15], which also treats the more delicate affine case. In part 3 we present a method to compute the Hilbert polynomial of Cohen-Macaulay schemes for which nice vanishing theorems are known for the dualizing module. The re-