The X-rank of a point p in projective space is the minimal number of points of an algebraic variety X whose linear span contains p. This notion is naturally submultiplicative under tensor product. We study geometric conditions that guarantee strict submultiplicativity. We prove that in the case of points of rank two, strict submultiplicativity is entirely characterized in terms of the trisecant lines to the variety. Moreover, we focus on the case of curves: we prove that for curves embedded in an even-dimensional projective space, there are always points for which strict submultiplicativity occurs, with the only exception of rational normal curves.

中文翻译：

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等级和边界等级严格次乘法的几何条件

投影空间中点p的X秩是线性范围包含p的代数变种X的最小点数。这个概念在张量积下自然是可乘的。我们研究了保证严格的可乘性的几何条件。我们证明，在第二点的情况下，严格的次乘性完全按照该品种的三割线来表征。此外，我们关注于曲线的情况：我们证明，对于嵌入在偶数维投影空间中的曲线，除了有理正态曲线外，总是存在发生严格次乘性的点。