5 problems

Let $E$ be a constructive exceptional bundle on $\mathbb{P}^3$, and let $h^i(E)$ denote the dimension of its $i$-th cohomology group, $\mu(E)$ its slope, and $\chi(E)$ its Euler ch…

Let $E$ be an exceptional vector bundle on $\mathbb P^2$, and let $\operatorname{Aff}_1(\mathbb Z)$ act naturally on vector bundles through $E\mapsto E^*$ and…

Let $e\geq 2$ and let $\mathcal{V}$ be an exceptional bundle on the Hirzebruch surface $\mathbb{F}_e$. Small-polarization stability conjecture. The bundle $\mathcal{V}$ is…

Let $E$ be a representative exceptional bundle on ${\mathbb P}^2$, meaning an exceptional vector bundle whose slope satisfies $0\leq\mu(E)\leq\frac{1}{2}$. Its rank is denoted by…

Let $Y$ be a smooth projective surface such that $K_Y$ is ample, $H^2(\mathcal T_Y)=0$, $H^{2,0}(Y)=0$, and $\pi_1(Y)=0$. Let $S$ be the set of slope vectors…