Partial elimination ideals and secant cones

TL;DR

This paper establishes a connection between partial elimination ideals and secant cones of algebraic schemes, providing a new computational approach and insights into fiber lengths of projections, supported by examples computed with Singular.

Contribution

It introduces a novel link between partial elimination ideals and secant cones, along with an algorithm for their computation and analysis of fiber lengths in projections.

Findings

Secant cones are defined by partial elimination ideals.

An algorithm for computing secant cones is developed.

Examples illustrate the theoretical results using Singular.

Abstract

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}.