Rank, Border Rank, Smoothable Rank and Cactus Rank
summary
Defines four concrete ranks for a point in P(W) relative to a variety X — usual rank, border rank, cactus rank (linear span of any finite subscheme) and smoothable rank (linear span of schemes that are limits of disjoint points). Proves inequalities and shows they coincide for r ≤ 2 and often r = 3 but fail beyond: gives an explicit S^3 C^5 example written as a limit of five distinct cubes with border rank 5, cactus and smoothable rank 6 and actual rank 9; tools: finite scheme/flat family language and ideal-theoretic descriptions in the symmetric algebra.
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