Geometric conditions for strict submultiplicativity of rank and border rank

The X -rank of a point p in projective space is the minimal number of points of an algebraic variety X whose linear span contains p . This notion is naturally submultiplicative under tensor product. We study geometric conditions that guarantee strict submultiplicativity. We prove that in the case of...

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Bibliographic Details
Published in:Annali di matematica pura ed applicata 2021-02, Vol.200 (1), p.187-210
Main Authors: Ballico, Edoardo, Bernardi, Alessandra, Gesmundo, Fulvio, Oneto, Alessandro, Ventura, Emanuele
Format: Article
Language:English
Subjects:
Mathematics
Mathematics and Statistics
Tensors
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Summary:The X -rank of a point p in projective space is the minimal number of points of an algebraic variety X whose linear span contains p . This notion is naturally submultiplicative under tensor product. We study geometric conditions that guarantee strict submultiplicativity. We prove that in the case of points of rank two, strict submultiplicativity is entirely characterized in terms of the trisecant lines to the variety. Moreover, we focus on the case of curves: we prove that for curves embedded in an even-dimensional projective space, there are always points for which strict submultiplicativity occurs, with the only exception of rational normal curves.
ISSN:0373-3114
1618-1891
DOI:10.1007/s10231-020-00991-6