5 problems

Let $X\subset\mathbb P^{2n}$ be a scroll of dimension $n$, and let $g$ denote the genus of its base curve. The genus conjecture for scrolls. Any such scroll has $g=0$ or $g=1$. Thi…

Scroll conjecture for extremal syzygies. If $\deg(L)\geq 2g+k$, then $res_C$ is surjective for $p=r(L)-k$, the largest value of $p$ for which…

Let $a$ be a positive integer, let $\overline{\mathbb{F}}$ be an algebraically closed field of characteristic $p$, and let $e=(e_1,e_2)\in\overline{\mathbb{F}}^2$ be general. Let…

Characteristic-zero purity conjecture. The union of scrolls $X_e(a,a)$ has a pure resolution unless $t_2/t_1\neq 1$ is a $k$-th root of unity for some $k\leq (a+1)/2$.

Let $X_e(a,b)$ denote the K3 union of scrolls considered in the paper, with parameters satisfying $a\geq b$. Optimal bound conjecture. The optimal bound ensuring the asserted resol…