There are close relations between tripartite tensors with bounded geometric ranks and linear determinantal varieties with bounded codimensions. We study linear determinantal varieties with bounded codimensions, and prove upper bounds of the dimensions of the ambient spaces. Using those results, we classify tensors with geometric rank 3, find upper bounds of multilinear ranks of primitive tensors with geometric rank 4, and prove the existence of such upper bounds in general. We extend results of tripartite tensors to n-part tensors, showing the equivalence between geometric rank 1 and partition rank 1.

翻译：具有边界几何等级的三方数母体和具有边界共振的线性决定性品种之间有着密切的关系。我们研究带有边界共振的线性决定性品种,并证明环境空间的尺寸具有上方界限。我们利用这些结果,对具有几何等级的数母体进行分类,将具有四等几何等级的原始数母体的多线性等级的上方界限进行分类,并证明存在这种总体的上方界限。我们把三方的数母体的结果扩大到 n-部分的数母体,显示1级和1级分区之间的等值。