TY - JOUR

T1 - Geometric conditions for strict submultiplicativity of rank and border rank

AU - Ballico, Edoardo

AU - Bernardi, Alessandra

AU - Gesmundo, Fulvio

AU - Oneto, Alessandro

AU - Ventura, Emanuele

PY - 2021

Y1 - 2021

N2 - 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.

AB - 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.

KW - Border rank

KW - Rank

KW - Secant variety

KW - Segre product

KW - Tensor product

UR - http://www.scopus.com/inward/record.url?scp=85085283807&partnerID=8YFLogxK

U2 - 10.1007/s10231-020-00991-6

DO - 10.1007/s10231-020-00991-6

M3 - Journal article

AN - SCOPUS:85085283807

VL - 200

SP - 187

EP - 210

JO - Annali di Matematica Pura ed Applicata

JF - Annali di Matematica Pura ed Applicata

SN - 0373-3114

IS - 1

ER -