13 problems

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…

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 $I({\rm GA}(t,k)')$ 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…

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 $R=K[x_1,\ldots,x_n]$, 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 $\Delta$ 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 $E$ be an exterior algebra and let $J\subset E$ be a pure ideal, meaning an ideal generated by products of linear forms. Suppose that $E/J$ satisfies property (): any two disti…

Let $S=\Bbbk[x_{1},\ldots,x_{n}]$, let $P=(x_{1}^{e_{1}},\ldots,x_{r}^{e_{r}})$, and set $R=S/P$. Let $I$ be a homogeneous ideal containing $P$, and let $L$ be the lex-plus-$P$ ide…

Let $S=\Bbbk[x_{1},\ldots,x_{n}]$, let $(f_{1},\ldots,f_{r})$ be a regular sequence with $\deg f_i=e_i$, and set … Let $I$ be the homogeneous ideal containing this regular sequence…

Let $S=\Bbbk[x_{1},\ldots,x_{n}]$, let $P=(x_{1}^{e_{1}},\ldots,x_{r}^{e_{r}})$, and let $F=(f_{1},\ldots,f_{r})$ be a homogeneous regular sequence with $\deg f_i=e_i$. Let $I$ be…