Strict inclusions of high rank loci

For a given projective variety X, the high rank loci are the closures of the sets of points whose X-rank is higher than the generic one. We show examples of strict inclusion arising from two consecutive high rank loci. Our first example comes from looking at the Veronese surface of plane quartics. Although Piene had already shown an example in which X is a curve, we construct infinitely many curves in P4 for which such strict inclusion appears. For space curves, we give two criteria to check whether the locus of points of maximum rank 3 is finite (possibly empty).