Partial elimination ideals and secant cones

Details Partial elimination ideals and secant cones Partial elimination ideals and secant cones

2010-01-20

arxiv.org/abs/1001.3592v2

Mathematics

Commutative Algebra

18 pages; revised version, to appear in Journal of Algebra

Scientific paper

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

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