29 problems

Let $I({\rm GA}(t,k)')$ be the edge ideal of the complement of the generalized Andrásfai graph ${\rm GA}(t,k)$, and let $\beta_{i,i+j}(I({\rm GA}(t,k)'))$ denote its graded Betti n…

Migliore–Miró-Roig conjecture. Such general equigenerated algebras have minimal Betti numbers among algebras with the Fröberg Hilbert function, subject to the constraints imposed b…

Let $A=R/I$ be a generic Artinian Gorenstein $K$-algebra of odd socle degree $s=2t-1\geqslant 3$ and embedding dimension $n\geqslant 4$. The ideal $I$ is the homogeneous defining i…

Let $R=K[x_1,\ldots,x_n]$, and let $I$ be a homogeneous ideal containing a regular sequence of degrees $2\leq a_1\leq\ldots\leq a_n$. Suppose that there exists a lex-plus-powers id…

Let $G$ be a graph. Write $J_G$ for its binomial edge ideal, let $\beta_{i,j}(J_G)$ denote its graded Betti numbers, and let $k_r(G)$ denote the number of $r$-cliques of $G$. Conje…

Let $A_{\mathbb{K}}$ be the polynomial ring over a ground field $\mathbb{K}$, and let $I$ be a $p$-stable ideal of $A_{\mathbb{K}}$. Write $\beta_{ij}(I)$ for its graded Betti numb…

Evans's lex-plus-powers conjecture. For all $i,j$,

Kalai–Kleinschmidt–Lee conjecture. For every $j$, the number of $j$-dimensional empty simplices of $P$ is at most the number of $j$-dimensional empty simplices of…

Lex Plus Powers conjecture for socles. Then

The Lex Plus Powers conjecture. If $\mathcal H$ is $\mathbb A$-lpp valid and $L_{\mathcal H,\mathbb A}$ is the $\mathbb A$-lex plus powers ideal attaining $\mathcal H$, then…

Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring with each $\deg x_i=1$, and let $\Delta$ be a simplicial complex on $[n]=\{1,\ldots,n\}$. Write $I_\Delta\subset S$ for its Stanley…

Let $\Delta$ be a simplicial complex on $[n]$, with Stanley–Reisner ideal $I_\Delta$. Let $\Delta^s$, $\Delta^e$, and $\Delta^c$ denote its symmetric algebraic, exterior algebraic,…

Let $S=K[x_1,\ldots,x_n]$ and let $I\subset S$ be an $\mathfrak m$-primary homogeneous ideal generated in degree $d$, where $\mathfrak m=(x_1,\ldots,x_n)$. Suppose that the minimal…

Let $X:f=0$ be a hypersurface in $\mathbb P^n$ with Jacobian algebra $M(f)$, and let the graded Betti numbers of a minimal resolution of $M(f)$ satisfy the relations in equations a…

Let $G$ be a closed graph, let $J_G$ be its binomial edge ideal, and let $\operatorname{in}_{\operatorname{lex}}(J_G^m)$ denote the lexicographic initial ideal of $J_G^m$. Write…

Let $C:f=0$ be a reduced plane curve that is not free, with $t(C)=\sum_{j=1}^{m-2}\epsilon_j\geq 1$, and let $\pi_C$ be the ordered partition of $t(C)$ whose parts are the positive…

Let $\Gamma\subseteq\mathbb{P}^{e}$ be a set of $d$ points in symmetric position. For nonnegative integers $k,m$, say that $\Gamma$ satisfies the property $\propa[k,m]$ if it is no…

Let $G$ be a graph on $m$ vertices and let $\alpha=(\alpha_1,\dots,\alpha_m)\in(\mathbb Z^+)^m$, where $G^\alpha$ is obtained by duplicating the vertices of $G$ according to…

Let $R=K[x_1,\ldots,x_n]$, and let $I$ be a homogeneous ideal containing a regular sequence of degrees $2\leq a_1\leq\ldots\leq a_n$. Let $L$ be a lex-plus-powers ideal associated…

Let $R$ be a polynomial ring over an infinite field, and let $I\subseteq R$ be a homogeneous ideal containing a regular sequence in prescribed degrees. For a fixed Hilbert function…

Let $\Delta$ be a $(d-1)$-dimensional balanced normal pseudomanifold, with $d\geq 4$, and suppose that $f_0(\Delta)=kd$ for an integer $k\geq 2$. Let $\mathbb{F}[\Delta]$ denote it…

Let a balanced normal pseudomanifold mean a balanced normal pseudomanifold, and let its Stanley–Reisner ring be denoted by $\mathbb{F}[\Delta]$. For a cross-polytopal stacked spher…

The surface Minimal Resolution Conjecture. The Betti table has the displayed shape

Lorenzini's Minimal Resolution Conjecture. The Betti table has only two nonzero rows, and in those rows

Let $G$ be a finite simple graph, and let $J_G$ be its binomial edge ideal. Write $\beta_{i,i+2}(J_G)$ for the graded Betti numbers in the linear strand of $J_G$, and let…