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On the Concept of k ‐Secant Order of a Variety
Author(s) -
Chiantini Luca,
Ciliberto Ciro
Publication year - 2006
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610706022630
Subject(s) - variety (cybernetics) , mathematics , dimension (graph theory) , order (exchange) , combinatorics , point (geometry) , secant method , pure mathematics , mathematical analysis , geometry , physics , newton's method , statistics , finance , nonlinear system , quantum mechanics , economics
For a variety X of dimension n in\sP r , r ⩾ n ( k + 1) + k , the k th secant order of X is the number μ k ( X ) of ( k + 1)‐secant k ‐spaces passing through a general point of the k th secant variety. We show that, if r > n ( k + 1) + k , then μ k ( X ) = 1 unless X is k ‐weakly defective. Furthermore we give a complete classification of surfaces X ⊂\sP r , r > 3 k + 2, for which μ k ( X ) > 1.