8 problems

Let $P_n=cmathbb{F}[x_1,\ldots,x_n]$, let $cell_1,cell_2$ be general linear forms in $P_n$, and let $a_{n,d}$ denote the number of lattice paths from $(0,0)$ to $(n+2-2s,n+2)$, with…

Migliore–Miró-Roig–Nagel conjecture. If $n=3$, then $R/I$ fails the WLP if and only if $d\geq 3$; if $n\geq 4$, then $R/I$ fails the WLP if and only if $d>1$. The conjecture concer…

Iarrobino's conjecture. Outside these exceptions, the algebras

Migliore, Miró-Roig and Nagel's conjecture. The ring $R/I$ fails the Weak Lefschetz property if and only if $d>1$. Furthermore, if $n=3$, then $R/I$ fails the Weak Lefschetz proper…

Let $K$ be a field of characteristic $0$, let $R=K[x_1,\dots,x_r]$, and let $\ell_1,\dots,\ell_n$ be general linear forms. For a positive integer $t$, set … and let $A=R/I$. Harbou…

Hochster–Seibert–Schenck conjecture. The quotient $R/I$ fails the WLP for all sufficiently large values of $t$.

Generic-powers WLP conjecture. If $n\ge r+1\ge 5$, then $S/I$ fails the weak Lefschetz property for all $t\gg0$.

Odd-variable equal-power WLP conjecture. The ring $R/I$ fails the weak Lefschetz property if and only if $d>1$. Furthermore, when $n=3$, so that $R$ has seven variables, $R/I$ fail…