Partial elimination ideals and secant cones

For any k∈N0, we show that the cone of (k+1)-secant lines of a closed subscheme Z⊆PKn over an algebraically closed field K running through a closed point p∈PKn is defined by the k-th partial elimination ideal of Z with respect to p. We use this fact to give an algorithm for computing secant cones. Also, we show that under certain conditions partial elimination ideals describe the length of the fibres of a multiple projection in a way similar to the way they do for simple projections. Finally, we study some examples illustrating these results, computed by means of Singular.