7 problems

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…

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

Ottaviani–Paoletti conjecture.