15 problems

Nonvanishing conjecture. If $A$ is not a principal ideal algebra, equivalently if it has at least one nonprincipal ideal, then

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.

Realizability conjecture. Every potential JDT arising for $s=6$ is realized by some pair $(A,\ell)$.

WLP conjecture. Every Artinian Gorenstein quotient of $k[x,y,z]$ has the WLP.

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 $K$ 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…

A standard graded Artinian algebra $A$ over a field $K$ has the strong Lefschetz property if there is a linear form $\ell\in A$ such that multiplication by every power $\ell^d$ ind…

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…

Let $n\geq 1$, let $l_1,\ldots,l_{2n+1}$ be nonconcurrent linear forms on $\check{\mathbb P}^2$, and let $f$ be a form of degree $2n+1$ on $\check{\mathbb P}^2$. Generalized Toglia…

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…

Seven-variable degree-three WLP conjecture. The ring $R/I$ fails the weak Lefschetz property (WLP). A computer calculation suggests failure, but the paper notes that a proof must j…

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…

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…