Geometric conditions for strict submultiplicativity of rank and border rank
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Geometric conditions for strict submultiplicativity of rank and border rank. / Ballico, Edoardo; Bernardi, Alessandra; Gesmundo, Fulvio; Oneto, Alessandro; Ventura, Emanuele.
In: Annali di Matematica Pura ed Applicata, Vol. 200, No. 1, 2021, p. 187-210.Research output: Contribution to journal › Journal article › Research › peer-review
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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 -
ID: 242663056