The Next Terracini Loci of Segre–Veronese Varieties and Their Maximal Weights

We describe all Terracini loci of Segre–Veronese varieties with at most roughly double the points of the minimal one. In this range we compute the maximum of all weights of the Terracini sets. To prove these results we use cohomological tools (residual exact sequences) applied to some critical schemes associated with a Terracini set and containing all of its points. We expect that these critical schemes will be a very useful tool for other related problems.