32 problems

Let $d\ge 3$, let $1\le a\le b\le d-1$ with $\operatorname{GCD}(a,b,d)=1$, and let … where $e$ is a $d$-th root of unity. Let $I:=I^d_{0,a,b}\subset R$ be the GT-system generated b…

Let $\Delta$ be a homology $(d-1)$-sphere over a field $\mathbf{k}$, with Stanley–Reisner ring $\mathbf{k}[\Delta]=S/I_\Delta$. An l.s.o.p. is a linear system of parameters…

WLP classification conjecture. Except when $(a_1,a_2,a_3,t)$ is $(2,9,13,9)$ or $(3,7,14,9)$, the quotient $P/I$ fails the weak Lefschetz property if and only if $t$ is even,…

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…

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…

Failure conjecture. For every $m\geq 1$, this multiplication map fails to have maximal rank; equivalently, the element $y_1y_3y_4\dots y_{m+3}\in A(w(B_m))_{m+2}$ is not in its ima…

GLN's conjecture. The displayed numerical condition, together with the requirement that at least two of $a,b,c$ are equal, describes all such algebras failing the weak Lefschetz pr…

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 $A=k[x_0,[...],x_m]/I$ be a graded artinian complete intersection $k$-algebra, where $I$ is generated by a regular sequence, and let $L$ be a general linear form. The Weak Lefs…

Let $I$ be generated by cubic monomials defining a basepoint-free map from $\mathbb{P}^r$ to projective space, and suppose its image is smooth. A map defined by cubic monomials to…

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…

Let $K$ be a Cohen–Macaulay simplicial complex, and let $\operatorname{sd}(K)$ denote its barycentric subdivision. The Kubitzke–Nevo conjecture. The simplicial complex…

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 different from $2$…

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…