13 problems

Let $S=k[x_1,[...truncated]$ be a polynomial ring, and let $S^{(c)}$ denote its $c$-th Veronese subring. In zero characteristic, the Green–Lazarsfeld index satisfies … For $n=3$, O…

Let $X$ be a projective variety, let $L$ be a very ample line bundle, and use the embedding $X\subset {\mathbb P}H^0(X,L)$. Assume that this embedding is $m$-regular, and let…

Let $r\geq 7$. For each $d$, consider the $d$-uple embedding of $\mathbb P^r$ and its Betti table. Unbounded torsion-prime conjecture. As $d\to\infty$, the number of primes $\ell$…

Let $r\geq 7$, and let $d$ be sufficiently large. Consider the $d$-uple embedding of $\mathbb P^r$ and its Betti table. Characteristic-dependence conjecture. For any $d\gg 0$, 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…

Last-row Schur module conjecture. The last nonzero syzygy space is

Let $K_{p,q}(\mathbb P^n,b;d)$ be the Koszul cohomology group of the $d$-uple Veronese embedding, let $\mathrm E_{p,q}(\mathbb P^n,b;d)$ be its monomial syzygy construction, and le…

For a Veronese embedding, let a row of its Betti table mean the sequence of Betti numbers with fixed homological or internal degree. Ein–Lazarsfeld normal-distribution conjecture.…

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 $K_{p,q}(n,b;d)$ be the Koszul cohomology groups associated with the $d$-th Veronese of $\boldsymbol{\rm P}^n$ and the twist $b$, and consider the vanishing range for these gro…

Let $K_{p,q}$ denote the Koszul cohomology groups governing the Betti table of a high-degree Veronese embedding, and let a row mean the collection of groups with fixed $q$. A Koszu…

Matching-complex homology conjecture. The only non-zero homology group of $C_{nd}^d$ for $n=1,\ldots,3d-1$ is

Let $\mathbf{P}^n$ be projective space, let $b\geq 0$, and let $K_{p,q}(\mathbf{P}^n,b;d)$ denote the relevant Koszul cohomology group for the $d$-uple Veronese embedding with coef…