10 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…

Let $cmathbb{F}$ be a field, let $cell_1,ldots,cell_{n+1}$ be general linear forms in $cmathbb{F}[x_1,ldots,x_n]$, and set … The Harima–Schreyer–Seceleanu conjecture. The algebra…

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…

Let $K$ be a field of characteristic $0$, let $R=K[x_1,\dots,x_r]$, and let $\ell_1,\dots,\ell_{r+1}$ be general linear forms. Set … and let $A=R/I$. Harbourne, Schenck and Secelea…

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…

Let $R=k[x_1,\dots,x_{2n+1}]$ be a polynomial ring in an odd number of variables, and let $I$ be an ideal generated by powers of general linear forms, all of the same degree $d$, w…