Cactus Varieties of Cubic Forms: Apolar Local Artinian Gorenstein Rings
summary
Proves a dimension formula for the cactus variety of cubic forms: for length r ≥ 17 the cactus and secant varieties have the same dimension (care needed for r > 13 because of non-smoothable local Gorenstein schemes); when r = 2n+2 the cactus fills ambient space and for 18 ≤ r < 2n+2 the cactus is strictly smaller. Uses polarity/apolarity, dehomogenization to local Artinian Gorenstein rings, and Hilbert function with symmetric decomposition to parameterize cubic tails and count dimensions; even r gives a unique local support.
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