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

Abstract

Let $C$ be a smooth projective curve of genus $g \geq 2$ over $\mathbb C$, and let $E^0$ be a vector bundle on $C$. We investigate the birational geometry of the Quot scheme ${\rm Quot}_C(E^0, k, n)$, which parametrizes quotients of $E^0$ of rank $k$ and degree $n$, and its fiber $\mathcal Q_L$ over ${\rm Pic}^n(C)$ for $n \gg 0$. Our main tool is the moduli space of stable pairs, which yields small $\mathbb Q$-factorial modifications (SQMs) of ${\rm Quot}_C(E^0, k, n)$ 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 ${\rm Quot}_C(E^0, k, n) \to {\rm Pic}^n(C)$ is a Mori dream morphism.