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Journal of the London Mathematical Society 2006 73(2):436-454; doi:10.1112/S0024610706022630
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© The London Mathematical Society

On the Concept of k-Secant Order of a Variety

Luca Chiantini and Ciro Ciliberto

Dipartimento di Scienze Matematiche e Informatiche, Università di Siena Pian dei Mantellini 44, 53100 Siena, Italy chiantini{at}unisi.it
Dipartimento di Matematica, Università di Roma Tor Vergata Via della Ricerca Scientifica, 00133 Roma, Italy cilibert{at}axp.mat.uniroma2.it

Received 7 July 2004. Revision received 14 June 2005.

For a variety X of dimension n in Pr, r ≥ n(k + 1) + k, the kth secant order of X is the number µk(X) of (k + 1)-secant k-spaces passing through a general point of the kth secant variety. We show that, if r > n(k + 1) + k, then µk(X) = 1 unless X is k-weakly defective. Furthermore we give a complete classification of surfaces X sub Pr, r > 3k + 2, for which µk(X) > 1.


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