On the \(X\)-ranks with respect to a low genus projective curve
type: Item
instance of: scholarly article
zbMATH Open document ID: 1265.14004
publication date: 2012-06-13T00:00:00Z
Mathematics Subject Classification ID: 14N05
Mathematics Subject Classification ID: 14P99
Mathematics Subject Classification ID: 14H52
zbMATH DE Number: 6045955
zbMATH Keywords: secant variety
zbMATH Keywords: rank with respect to a variety
zbMATH Keywords: border rank
zbMATH Keywords: real curve
zbMATH Keywords: elliptic curve
author: E. Ballico
MaRDI profile type: Publication
title: On the \(X\)-ranks with respect to a low genus projective curve
published in: Mathematical Reports
review text: Let \(X\subset\mathbb P^n=\mathbb P^n_\mathbb C\) be a nonsingular, non-degenerate, complex projective curve. For \(r\geq 2\), let \(\text{Sec}_r(X)\) denote the \(r\)th secant variety of \(X\), that is, the closure of the union of the \((r-1)\)-planes in \(\mathbb P^n\) spanned by \(r\) points of \(X\). It is known that \(\dim\text{Sec}_r(X)=\text{min}(n,2r-1)\), hence if \(\rho :=\lfloor (n+2)/2\rfloor\) then \(\text{Sec}_\rho(X)=\mathbb P^n\) and \(\text{Sec}_{\rho -1}(X)\neq\mathbb P^n\). Put \(U:=\mathbb P^n\setminus\text{Sec}_{\rho -1}(X)\) and, for \(P\in U\), let \(\mathcal S(X,P)\) denote the set of the subsets \(S\) of \(X\) consisting of \(\rho\) elements such that \(P\) belongs to the linear span of \(S\).NEWLINENEWLINE Assume, from now on, that \(X\) is a ``real'' curve and that \(X(\mathbb R)\neq\emptyset\). The complex conjugation \(\sigma:\mathbb P^n(\mathbb C)\rightarrow\mathbb P^n(\mathbb C)\) maps \(X(\mathbb C)\) to \(X(\mathbb C)\). If \(P\in U\cap\mathbb P^n(\mathbb R)\) then \(\sigma\) induces a map \(\mathcal S(X,P)\rightarrow\mathcal S(X,P)\). The author defines the ``signature'' of \(P\) to be the following set of integers: NEWLINE\[NEWLINE\tau(P):=\{\text{card}(S\cap X(\mathbb R))\, | \, S\in\mathcal S(X,P),\, \sigma(S)=S\}\, .NEWLINE\]NEWLINE The main results of the paper under review are the following ones:NEWLINENEWLINE 1) For every \(0\leq t\leq\rho\) with \(t\equiv\rho\) (mod 2), there exists a nonempty open subset \(U_t\) of \(U\cap\mathbb P^n(\mathbb R)\) such that \(\tau(P)=\{t\}\), \(\forall\, P \in U_t\);NEWLINENEWLINE 2) If \(n=2k+1\) and \(X\) is a rational normal curve defined over \(\mathbb R\) then \(\tau(P)\) has only one element, \(\forall\, P \in U\cap\mathbb P^n(\mathbb R)\);NEWLINENEWLINE 3) If \(n=2k+1\) and \(X\) is a real, linearly normal embedding of an elliptic curve then, \(\forall\, P \in U\cap\mathbb P^n(\mathbb R)\) such that \(\mathcal O_X(2Z)\) is not isomorphic to \(\mathcal O_X(1)\) for each \(Z\in\mathcal …
reviewed by: Iustin Coandă
description: scientific article; zbMATH DE number 6045955