4 problems

Let $F\in K[x_0,\ldots,x_n]$ be a general cubic form, and let its cactus rank be the minimal length of a finite apolar subscheme, allowing singularities. In fact, its cactus rank i…

Let $q$ be the quadric and let $N(n,r)=\binom{n+r}{n}$ be the cactus rank of $q^r$. Denote by $\operatorname{VAPS}_G(q^r,N(n,r))$ the variety of smoothable apolar schemes of length…

Let $m=x_0^{a_0}\cdots x_n^{a_n}$ be a monomial. Write $d=\sum_{i=0}^n a_i$, and let $\underline{\mathrm{cR}}_{d}(m)$ and $\underline{\mathrm{R}}_{d}(m)$ denote the border cactus r…

Hilbert-function conjecture. The Hilbert function of a minimal local Artinian Gorenstein scheme for a general cubic form with local cactus rank $l$ has the displayed form according…