9 problems

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

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 $K$ 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…

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 $\mathbb{F}$ be a field and let $R=\mathbb{F}[x_1,\ldots,x_n]$ have its standard grading. A complete intersection algebra is a quotient $R/\langle f_1,\ldots,f_n\rangle$ by a h…

Let $A$ be an artinian algebra, and let … be inclusions of subcategories satisfying (G3). Suppose that $n\geq 0$ and that either (G1) holds and every $M\in\mathcal{C}$ admits an ep…

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, and consider the $h$-vectors of quadratic Artinian $K$-algebras and the $f$-vectors of simplicial complexes. Eisenbud–Green–Harris conjecture. Every $h$-vector…