On the \(X\)-ranks with respect to a low genus projective curve (Q2891131)

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 \(Z\in\mathcal S(X,P)\), \(\tau(P)\) consists of at most two elements and, moreover, \(\forall\, 0\leq t_1\leq t_2\leq k+1\, (=\rho)\) with \(t_1\equiv t_2\equiv k+1\) (mod 2), there exists such a \(P\) with \(\tau(P)=\{t_1,t_2\}\).