28 problems

Conjecture. Every standard graded Artinian Gorenstein algebra of embedding dimension three over a field of characteristic zero has the weak Lefschetz property.

Let $C\subset\mathbb P^3$ be the union of $r$ general lines. The strong Lefschetz property conjecture. Then $M(C)$ has SLP. Here $M(C)$ denotes the Hartshorne–Rao module of $C$, an…

Ragunathan–Van Tuyl's conjecture. Fix an integer $k\geq 3$. If $k$ is odd, then there exists an $m$ such that $A(n,k)$ fails to have the WLP for every $n\geq m$. If $k$ is even, th…

Recursive description of the integers $c_j$. If $b-j\leq rq$, then $c_j(a,b)=rq$. Otherwise,

Let $k$ be a field of arbitrary characteristic and let $\Delta$ be a connected oriented simplicial pseudomanifold over $k$ of dimension $d-1$ with vertex set $V$. Let $\ell$ be the…

Almkvist–Lefschetz strengthening. (i) For each $m$, the algebra $A(m,n)$ satisfies the SLP for all $n\geq 11$. (ii) Moreover, if $m$ is even, then $A(m,n)$ satisfies the SLP for al…

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…

Let $R=k[x_1,\dots,x_n]$ where $k$ is a field of characteristic zero. Fix $n,d\geq 3$ and let $\mu$ lie in the interval $\left[n,\binom{n+d-1}{d}\right]$. A monomial ideal $I$ mini…

Let $A$ be an Artin Gorenstein ring with codimension $c$ and regularity $r$, and let a Betti table refer to the graded Betti table of a minimal free resolution of $A$. The codimens…

Let $A$ be an Artinian Gorenstein algebra of codimension three, with socle degree $j_A$, over an infinite field $\mathsf{k}$ of characteristic $p>j_A$. For a general enough linear…

Let $S=\mathbb{C}[x_0,\ldots,x_n]$ be the graded polynomial ring in $n+1$ variables, with $n\geq 2$, and let $f\in S_d$ be a homogeneous polynomial such that the hypersurface…

Let $a\geq b\geq c$, set $d=a+b+c$, and define … The weak Lefschetz failure conjecture. The algebra $R_{a,b,c}$ fails the weak Lefschetz property if and only if both of the followi…

SLP classification conjecture. The equigenerated monomial almost complete intersections of codimension three with the SLP are precisely those given by

The algebraic $g$-conjecture. Every rational homology sphere has Lefschetz property.

Fix positive integers $m,n$, let $W=G(m,1,n)$ and $K=\mathfrak{S}_n\subset W$, and define … Here $e_i$ is the $i$-th elementary symmetric function in $x_1,\ldots,x_n$, and…

Let $\Delta$ be an orientable simplicial $\mathbb{R}$-homology $d$-manifold without boundary, with $H_1(\Delta;\mathbb{Z}/2\mathbb{Z})=0$ whenever $d\geq2$. Let $\Upsilon$ be a sub…

Let $A$ be the Artinian algebra and let $\times L^b$ denote multiplication by the $b$-th power of a general linear form as in Corollary. The corollary currently assumes $b\leq t$.…

Let $R=k[x,y,z]$ and let $A=R/I$ be any Artinian quotient generated by powers of general linear forms. For a general linear form $L$, multiplication by $L^2$ is the graded map … fo…

Let $R=k[x,y,z]$ and let … where $L_1,\dots,L_4$ are general linear forms. Levelness conjecture. The quotient $R/I$ is level with Cohen–Macaulay type $k$. This predicts a precise s…

Bjorner--Swartz conjecture. Every doubly Cohen--Macaulay simplicial complex has the dual WLP.

Colored SLP conjecture. Any balanced simplicial sphere, or at least any balanced simplicial polytope, has the colored SLP over a field of characteristic $0$.

Let $A=\bigoplus_i A_i$ be an Artinian Gorenstein algebra presented by quadrics over a field $\mathbb{K}$ of characteristic zero. The Weak Lefschetz Property means that there exist…

Let $A$ be an Artinian Gorenstein algebra of socle degree at least three, presented by quadrics over a field $\mathbb{K}$ of characteristic zero. For $L\in A_1$, write…

Let $K$ be a field and let $R=K[x,y,z]$. An Artinian monomial ideal $I\subset R$ gives an Artinian monomial algebra $R/I$, whose type is the dimension of its socle. The algebra…

Let $I_{(1)},\ldots,I_{(\ell)}$ be the tangent-space ideals at $\ell$ general points of the variety of reducible hypersurfaces, let $B$ be the coordinate ring of the join of the sc…