3 problems

Let $C$ be a nonhyperelliptic curve of genus $g\geq 9$. Let $E\to C$ be a vector bundle of rank $r$ and slope at most $g-2$ such that the tautological line bundle…

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 $k\leq (a+1)/2$.