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

We here develop some new algorithms for computing several invariants attached to a projective scheme (dimension, Hilbert polynomial, unmixed part,… ) that are based on liaison theory and therefore connected to properties of the canonical module. The main features of these algorithms are their simplicity and the fact that their complexity is controlled by the complexity of the output, when it exceeds a linear function on the degrees of the input. We also give bounds for the Castelnuovo-Mumford regularity (which controls the complexity of the output) in low dimension, and give a reasonable algorithm to check smoothness of the unmixed part (one case where good bounds are known for complexity).