6 problems

Let $S$ be a polynomial ring over $\mathbb C$, and let $g_1,\ldots,g_r$ be generic forms of degree $d>1$. Nicklasson's conjecture. The ideal … has the same Hilbert series as an ide…

Let $S$ be a polynomial ring in $n$ variables over $\mathbb C$, and let $\ell_1,\ldots,\ell_r$ be generic linear forms. For a positive integer $d$, set…

Graded-piece characterization conjecture. One has $R_{i,\mathbf{h}}\neq 0$ if and only if $\mathbf{h}\in\mathcal{H}_j$; moreover, for $\mathbf{h}\in\mathcal{H}_j$,

Let $\mu_{i,\mathbf{h}}$ be the maps defined in Definition, indexed by $i$ and a vector $\mathbf{h}$, and let $\mathrm{wt}(\mathbf{h})$ denote the weight of $\mathbf{h}$. Let $k$,…

Let $R_{2,n,d}$ be the graded algebra under consideration, with socle ${\rm Soc}(R_{2,n,d})$ and graded component $[R_{2,n,d}]_{2d-2}$. Levelness conjecture. $R_{2,n,d}$ is a level…

Let $R=k[x_1,\ldots,x_n]$ be a polynomial ring over a field $k$ of characteristic $0$ or sufficiently large. For a linear degree function $\phi$ satisfying … let $I_\phi$ be the mo…