Given a closed subvariety X in a projective space, the rank with respect to X of a point p in this projective space is the least integerr such that p lies in the linear span of some r points of X. Let Wk be the closure of the set of points of rank with respect to X equal to k. For small values of k such loci are called secant varieties. This article studies the loci Wk for values of k larger than the generic rank. We show they are nested, we bound their dimensions, and we estimate the maximal possible rank with respect to X in special cases, including when X is a homogeneous space or a curve. The theory is illustrated by numerous examples, including Veronese varieties, the Segre product of dimensions (1, 3, 3), and curves. An intermediate result provides a lower bound on the dimension of any GLn orbit of a homogeneous form.

On the locus of points of high rank

Mella, Massimiliano;
2018

Abstract

Given a closed subvariety X in a projective space, the rank with respect to X of a point p in this projective space is the least integerr such that p lies in the linear span of some r points of X. Let Wk be the closure of the set of points of rank with respect to X equal to k. For small values of k such loci are called secant varieties. This article studies the loci Wk for values of k larger than the generic rank. We show they are nested, we bound their dimensions, and we estimate the maximal possible rank with respect to X in special cases, including when X is a homogeneous space or a curve. The theory is illustrated by numerous examples, including Veronese varieties, the Segre product of dimensions (1, 3, 3), and curves. An intermediate result provides a lower bound on the dimension of any GLn orbit of a homogeneous form.
Buczyński, Jarosław; Han, Kangjin; Mella, Massimiliano; Teitler, Zach
File in questo prodotto:
File Dimensione Formato  
On the locus of points of high rank.pdf

accesso aperto

Tipologia: Full text (versione editoriale)
Licenza: Creative commons
Dimensione 543.94 kB
Formato Adobe PDF
543.94 kB Adobe PDF Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11392/2380932
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 15
  • ???jsp.display-item.citation.isi??? 13
social impact