24 problems

Link-existence conjecture. For any $i$, $3\leq i\leq d-2$, if the $i$-th link does not exist, so that there is a common factor in the initial degree of the ideal of $Y_{i-1}$, then…

Exact-one failure conjecture. If there is a complete intersection $J\in CI(d,d,d,d)$ that fails WLP by more than one, then there exists a complete intersection $I\in CI(d,d,d,d)$ t…

Let $R=K[x,y,z]$, and let … be a level Artinian almost complete intersection, where $t>0$ and $0\leq\alpha\leq\beta\leq\gamma$. Assume that $K$ is algebraically closed of character…

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…

Non-square grid conjecture. The Artinian algebra $A$ fails to have the weak Lefschetz property.

Let $S$ be a polynomial ring in $n$ variables, let $I\sim\mathcal{I}(n,D,p)$, and consider the random Artinian algebra … where $n\geq 3$. Set $p=1/D^2$. The vanishing WLP probabili…

Iterated Weak Lefschetz Property conjecture. For any linear forms $\ell_1,\dots,\ell_i$ in $R$, the quotient $R_i$ enjoys the Weak Lefschetz Property.

Harbourne–Schenck–Seceleanu conjecture. If $r+1\geq n\geq5$, then $R_{n,r,d}$ fails the WLP for $d\gg0$.

Let $K$ be a field of characteristic zero, let $R=K[x_0,\ldots,x_n]$, and let $L\in R$ be a general linear form. Consider the ideal … The weak Lefschetz property (WLP) for an Artin…

Let $R$ be the polynomial ring in $r$ variables over a field of positive characteristic, and let $I_{r,k,d}$ be the ideal … Consider the Artinian algebra $R/I_{r,k,d}$. Weak Lefsch…

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…

Characteristic bound conjecture. Then $R/I$ has the weak Lefschetz property in every characteristic $p>\frac{1}{2}(a+b+c)$.

The odd axial-puncture criterion.

Let $\Delta$ be a connected orientable homology $(d-1)$-manifold without boundary whose vertex links have the weak Lefschetz property. Let $\Theta$ and $w$ be as in Theorem 5.4, se…

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

Let $R/I$ be an artinian Gorenstein algebra presented by quadrics, with socle degree at least $3$, and suppose that $R$ is defined over a field of characteristic zero. The WLP conj…

Let $R$ be the polynomial ring in the variables used to define the ideal, over a field $K$ of characteristic zero. Let $0<\alpha\leq\beta\leq\gamma\leq2(\alpha+\beta)$, let…

Let $a=b$ and $\alpha=\beta$, so the punctured hexagon $H_{a,a,c,\alpha,\alpha,\gamma}$ is symmetric. Let $A$, $C$, and $M$ be the parameters associated with this hexagon, let…

Let $A_{r,2}$ denote the quotient algebra formed from $r$ generic linear forms in the construction used in the paper, with the generators squared, and let WLP denote the weak Lefsc…

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 $K$ be a field of characteristic $2$, let $d$ be a positive integer, and set … The characteristic-two conjecture. The algebra $A$ has the weak Lefschetz property if and only if…

Let $K$ be a field of characteristic zero, let $R=K[x,y,z]$, and let $I_{\alpha,\beta,\gamma,t}$ denote the level monomial almost complete intersection used in the preceding criterion…