We introduce the concise secant varieties, which are, informally speaking, modular partial desingularisations of secant varieties to Segre embeddings. More precisely, they are projective and birational to the abstract secant varieties, yet each of their points corresponds to a concise tensor of appropriate border rank (that is, to a minimal border rank tensor). We discuss implications throughout the theory of tensors, including a characterisation of border rank $\leq r$ tensors as unrestrictions of minimal border rank $r$ tensors (also in the Veronese and Segre-Veronese cases), a characterisation of tensors with cactus rank $\leq r$, concise versions of border apolarity including the fixed point theorem, concise Varieties of Sums of Powers, counting points on the second secant variety, connections to defectivity and identifiability in the Segre case, to the Salmon conjecture etc.

翻译：我们引入了简洁割线簇，其非正式定义是将塞格雷嵌入的割线簇进行模性部分去奇异化。更精确地说，它们是抽象割线簇的射影双有理变换，且每个点对应一个具有恰当边界秩的简洁张量（即最小边界秩张量）。我们讨论了这一概念在整个张量理论中的影响，包括将边界秩≤r的张量刻画为最小边界秩r张量的无约束形式（同样适用于韦罗内塞与塞格雷-韦罗内塞情形）、对仙人掌秩≤r张量的刻画、包含不动点定理的边界极性理论的简洁版本、简洁的幂和簇变体、第二割线簇上的计数点、与塞格雷情形下的缺陷性和可辨识性的关联，以及对萨尔蒙猜想等方面的应用。