9 problems

Let $S=\mathbb{C}[x_1,\ldots,x_n]$ with $n\geq 3$, and let $f_j\in S_{d_j}$ be homogeneous polynomials of degrees $d_j\geq 2$ such that … is a zero-dimensional complete intersectio…

The strong full Perazzo conjecture. The Hilbert vector $H$ is strongly minimal in $\mathcal{AG}(r,d)$.

The full Perazzo conjecture. The Hilbert vector $H$ is minimal in $\mathcal{AG}(r,d)$; equivalently, if $\hat{H}\preceq H$ is a comparable Artinian Gorenstein Hilbert vector, then…

Let $A$ be an Artinian Gorenstein algebra of codimension three, with socle degree $j_A$, over an infinite field $\mathsf{k}$ of characteristic $p>j_A$. For a general enough linear…

Let $M$) be an upper triangular matrix of size $d+1$ with non-negative entries. For an upper triangular matrix, let $\operatorname{diag}(i,M)$ denote its $i$-th diagonal vector, an…

Let $A=\bigoplus_i A_i$ be an Artinian Gorenstein algebra presented by quadrics over a field $\mathbb{K}$ of characteristic zero. The Weak Lefschetz Property means that there exist…

Let $A$ be an Artinian Gorenstein algebra of socle degree at least three, presented by quadrics over a field $\mathbb{K}$ of characteristic zero. For $L\in A_1$, write…

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

Associated-form conjecture. For any ${\tt I}\in{\mathcal I}_n^m$ there exists an absolute invariant $\mathbf I$ of forms of degree $n(m-2)$ on $\mathbb C^n$ such that, for all…