14 problems

Let $M$ be the coordinate ring of the Veronese, under the hypotheses of Theorem, and let $\beta_{i,i+1}$ denote the corresponding graded Betti numbers. Erman–Martinova conjecture.…

Unimodality conjecture. If either $d_1$ or $d_2$ is sufficiently large, then each of the following functions is unimodal:

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…

For the $d$-uple Veronese embedding of $\mathbb P^2$, let $\beta(\mathbb P^2,b;d)$ denote its Betti table with twist parameter $b$, and let its Boij–Söderberg coefficients be the c…

Recursive-coefficient conjecture. For $\varepsilon<s<\varepsilon+c$, one has

Let $k$ be a row index, and consider a Betti table for which Theorem yields nonvanishing Betti numbers in row $k$. The $k$th row is viewed as a sequence of Betti numbers indexed by…

Pardue's conjecture. The $\mathbb{N}$-graded Betti table of $I(S\otimes_{\mathbb Z}\ell)$ is independent of $\operatorname{char}\ell$ (equivalently, of $\ell$) for all fields…

Let $G$ be a graph formed by gluing $k$ copies of the four-cycle $C_{4}$ together along one edge. Write $b_{i,j}(\overline{G})$ for the graded Betti numbers of its graph curve. Bet…

Let $G$ be a graph and let $\overline{G}\subset \mathbb{P}^{n}$ be its graph curve. Write $b_{i,j}(\overline{G})$ for the graded Betti numbers, and call an edge of $G$ a bridge if…

Let $G$ be a graph with $d$ vertices, genus $g$, and girth $b3$. Let $\overline{G}$ be the corresponding graph curve, with Betti numbers $b_{i,j}(\overline{G})$. Burnham–Rosen–Sidm…

Extremal Betti table generation conjecture. All extremal Betti tables of the cone of bigraded $T$-modules with finite length are generated by Betti tables of modules $M$ that satis…

Let $X$ be a smooth projective variety of dimension $n$, let $L_d$ be the very positive line bundles used to define the Betti numbers $k_{p,q}(X;L_d)$, and write $r_d$ for the rele…

Let $I\subseteq R$ be an equigenerated homogeneous ideal generated in degree $r$. For each pair of indices $(i,j)$, consider the graded Betti numbers $\beta_{i,j+dr}(I^d)$ as $d$ v…

Let $I_G=(m_0,m_1,\ldots,m_k)\subseteq \Bbbk[x_1,\ldots,x_N]$ be a square-free monomial ideal with … For a homogeneous ideal $I$ equigenerated in degree $r$, let…