Entry loci and ranks

We study entry loci of varieties and their irreducibility from the perspective of X-ranks with respect to a projective variety X. These loci are the closures of the points that appear in an X-rank decomposition of a general point in the ambient space. We look at entry loci of low degree normal surfaces in P4 using Segre points of curves; the smooth case was classically studied by Franchetta. We introduce a class of varieties whose generic rank coincides with the one of its general entry locus, and show that any smooth and irreducible projective variety admits an embedding with this property.