Partial elimination ideals and secant cones

For any $k\in \text{Nat}$, we show that the cone of $(k+1)$-secant lines of a closed subscheme $Z\subset {\mathbb{P}}_K^n$ over an algebraically closed field $K$ running through a closed point $p\in {\mathbb{P}}_K^n$ 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 {\sc Singular}.