5 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 $S=k[\alpha_1,\ldots,\alpha_n]$ and $P=k[x_1,\ldots,x_n]$ be polynomial rings over a field $k$, with $S$ acting on $P$ by contraction. For a homogeneous polynomial $f\in P$, wr…

Let $d\geq 3$ be an integer, let $F$ be a generic ternary degree $d$ form, and let $\mathbb{X}(d+1)$ denote a star configuration of $d+1$ points in the projective plane. Existence…

Canonical-grading conjecture. A general algebra of type $(n,d)$ is canonically graded if and only if $(n,d)$ belongs to the following list: