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Equations defining secant varieties: geometry and computation. (English) Zbl 1251.14043

Fløystad, Gunnar (ed.) et al., Combinatorial aspects of commutative algebra and algebraic geometry. The Abel symposium 2009. Proceedings of the 6th Abel symposium, Voss, Norway, June 1–4, 2009. Berlin: Springer (ISBN 978-3-642-19491-7/hbk; 978-3-642-19492-4/ebook). Abel Symposia 6, 155-174 (2011).
The \(k\)th secant variety \(\Sigma_k\) of a variety \(X \subset \mathbb{P}^n\) is the Zariski closure of the union of \(k\)-planes in \(\mathbb{P}^n\) that meet \(X\) in at least \(k+1\) points. In this article, the authors study secant varieties of curves with emphasis on concrete examples and calculations.
In Section 2 they show how to compute the ideals of secant varieties via elimination and prolongation. For both approaches, they provide algorithms, concrete examples and Macaulay 2 source code.
In Section 3 they consider the desingularization of secant varieties. It is not their goal to repeat the general theory but to illustrate it at hand of explicit examples. Moreover, they show how cohomology can be used to prove that the secant variety is projectively normal.
For the entire collection see [Zbl 1215.13001].

MSC:

14N05 Projective techniques in algebraic geometry
13D02 Syzygies, resolutions, complexes and commutative rings
14H99 Curves in algebraic geometry
14Q05 Computational aspects of algebraic curves

Software:

Macaulay2
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