12 problems

Let $K$ be a field, let $R=K[x_1,\dots,x_N]$ be a polynomial ring, and let $I$ be an ideal generated by $n$ forms of degrees $d_1,\dots,d_n$. Write $\operatorname{pd}_R(I)$ for the…

Eliahou–Villarreal conjecture. If the edge ideal of a connected graph $G$ has a 2-linear resolution, then

Let $P(n,d)$ denote a bound for the projective dimension in Stillman's question: for every polynomial ring $S$ over a field and every graded ideal generated by $n$ forms of degree…

Let $S$ be a polynomial ring over a field, and let $I\subseteq S$ be an ideal generated by $n$ quadrics and of height $h$. Ananyan–Hochster's quadratic bound conjecture. One has ……

Let $n_1$ be a positive integer, and let $R_{n_1,1}$ be the ring and $I_{L_2(n_1,1)}$ the ideal appearing in the paper, with $\mathrm{in}_{\prec}(I_{L_2(n_1,1)})$ denoting its init…

Draisma–Eggermont–Núñez-Betancourt–Römer conjecture. The projective dimension $\operatorname{pdim}(I_m)$ is given by a linear function of $m$ for $m\gg 0$. The analogous eventual l…

Monotonicity conjecture. Let $r,n\geq 2$ be arbitrary integers. If $(p,r)\in\operatorname{pdreg}(n)$, then $(p,r-1)\in\operatorname{pdreg}(n)$. The preceding results show that the…

Polynomial Stillman-bound conjecture. For all characteristics, the projective dimension of $R/I$ is at worst

Let $R$ be a polynomial ring over a field, and let $I$ be an ideal generated by three forms of degree three. Write $\mathop{\mathrm{pd}}\nolimits(R/I)$ for the projective dimension…

Let $M$ be a monomial ideal minimally generated by more than one monomial. Suppose that some minimal generator divides the least common multiple of every pair of generators. Projec…

Converse characterization conjecture. Then $\operatorname{pd}(\mathcal{H})=|V(\mathcal{H})|=2$ if and only if $\mathcal{H}$ satisfies $(\sharp)$. The authors are asking whether thi…

Let $I$ be an ideal with $N$ generators, whose degrees are $d_1,\ldots,d_N$. Zhang's conjecture. An upper bound for the projective dimension of $I$ is … This conjecture proposes a…