12 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 $\mathsf R$ be the homogeneous coordinate ring of $\mathbb P^n$, let $r\geq 0$, and let $L_1,\dots,L_t$ be generic linear forms. Consider the ideal … The Iarrobino conjecture.…

Let $\mathsf R$ be the homogeneous coordinate ring of $\mathbb P^2$, let $r\geq 0$, let $t>8$, and let $L_1,\dots,L_t$ be generic linear forms. Set … For a graded quotient, denote…

Let $\mathsf R$ be the homogeneous coordinate ring of $\mathbb P^2$, let $t\geq 9$, let $r\geq 0$, and let $L_1,\dots,L_t$ be linear forms. Write $H(\mathsf R/I,k)$ for the Hilbert…

Let $R=k[x_1,\ldots,x_n]$ be a polynomial ring over a field $k$ of characteristic $0$ or sufficiently large. Let $J_\phi$ be the ideal generated by…

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…

Holtz–Ron conjecture. The power ideal $C_{\mathcal A,-2}$ is generated by $\mathcal A$-monomials.

Holtz–Ron conjecture. The Main Theorem holds in the internal case, that is, for $k=-1$ and $J=\{X\}$.