3 problems

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…

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…

Let $F\in k[x_0,\ldots,x_n]$ be a generic homogeneous cubic form, and let $cr(F)$ denote its cactus rank and $r(F)$ its rank. Cactus-rank conjecture. The cactus rank equals the ran…