Entry loci and ranks (Q2038406)

Fix a non-degenerate projective variety \(X\subset\mathbb P^n \) and a point \(q\in\mathbb P^n\). The \(X\)-rank of \(q\) with respect to \(X\) is the minimal integer \(r_X\) such that \(q\) lies in the span of \(r_X\) points of \(X\). The entry locus \(\Gamma_q(X)\) of \(q\) with respect to \(X\) is the closure of the union of all subsets of cardinality \(r_X\) of \(X\) whose span contains \(q\). The authors study some geometric properties of the entry loci. In particular, the authors describe the Segre locus \(\mathfrak S(\Gamma_q(X))\), i.e. the locus of points \(p\) such that the projection of \(\Gamma_q(X)\) from \(p\) is not birational. The description allows the authors to characterize the general entry locus of surfaces \(X\) in \(\mathbb P^4\) with isolated singularities. The authors also find a criterion for the irreducibility of entry loci of some projections of varieties.