25 problems

Let an extremal ray of the cone of Betti tables be given, and consider its integral points. Eisenbud–Fløystad–Weyman conjecture. Every sufficiently large integral point on an extre…

Let $S$ be the polynomial ring in the paper, and let $d=(d_0,\dots,d_{n+1})\in\mathbb{Z}^{n+2}$ be a strictly increasing degree sequence. A pure resolution of type $d$ is a minimal…

Let $R$ be the quotient ring defined by $n+1$ general quadrics in $n$ variables, with $n\geq 5$ odd, and write $\ell$ for the parameter governing the strands in its minimal graded…

For each degree, choose the lex-leading monomial and use it to produce nonzero Betti entries in the corresponding row of the Betti table. Ein–Erman–Lazarsfeld conjecture. Choosing…

Let $M$ be a Veronese module, let $\beta_{i,j}(M)$ denote its graded Betti numbers, and consider the cited range of entries in a given row. Ein–Erman–Lazarsfeld conjecture. All Bet…

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.…

Let $I$ be a graded ideal, let $\mathfrak m$ denote the homogeneous maximal ideal, and let $c_I$ and $W_{c_I}$ be the quantities defined in the paper. Then the regularity stabiliza…

Let $A$ be an Artin Gorenstein algebra with Hilbert vector $H(A)=(1,4,7,7,4,1)$, and let the Betti tables below refer to the graded Betti tables of such algebras. Nine-table and st…

Let $A$ be an Artin Gorenstein algebra with Hilbert vector $H(A)=(1,4,4,4,4,1)$. A Betti table means the graded Betti table of a minimal free resolution of $A$. Uniqueness and stro…

The stabilization conjecture for Betti tables. The conjectured Betti tables for $r=2$ and $r=3$ are the displayed tables in the source, and the observed patterns suggest that the e…

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…

Let $X$ be a smooth projective variety, and consider increasingly positive Veronese embeddings of $X$. The ranks of the syzygy modules are understood after an appropriate rescaling…

Let $B$ be the homogeneous coordinate ring of a rational normal curve, and let $\mathbb{B}(B)$ denote the cone of Betti tables of finitely generated $B$-modules. For each $c=0,1,2$…

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…

Let $R=k[x_1,\ldots,x_n]$ be a polynomial ring over a field $k$. The Betti tables of graded minimal free resolutions of Cohen–Macaulay $R$-modules form a monoid under pointwise add…