Birational Geometry of Quot Schemes on smooth projective curves via Stable Pairs

Let $C$ be a smooth projective curve of genus $g\ge 2$ over $\mathbb{C}$, and let $E^0$ be a vector bundle on $C$. We investigate the birational geometry of the Quot scheme ${\mathrm{Q}\mathrm{u}\mathrm{o}\mathrm{t}}_C(E^0,k,n)$, which parametrizes quotients of $E^0$ of rank $k$ and degree $n$, and its fiber ${\mathcal{Q}}_L$ over ${\mathrm{P}\mathrm{i}\mathrm{c}}^n(C)$ for $n\gg 0$. Our main tool is the moduli space of stable pairs, which yields small $\mathbb{Q}$-factorial modifications (SQMs) of ${\mathrm{Q}\mathrm{u}\mathrm{o}\mathrm{t}}_C(E^0,k,n)$ and ${\mathcal{Q}}_L$. We explicitly describe the nef, movable, and effective cones of each SQM. Consequently, we prove that ${\mathcal{Q}}_L$ is a Mori dream space and that the determinant morphism ${\mathrm{Q}\mathrm{u}\mathrm{o}\mathrm{t}}_C(E^0,k,n)\to {\mathrm{P}\mathrm{i}\mathrm{c}}^n(C)$ is a Mori dream morphism.