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

TL;DR

This paper studies the birational geometry of Quot schemes on smooth projective curves using stable pairs, showing they are Mori dream spaces with explicitly described cones.

Contribution

It introduces the use of stable pairs to construct SQMs of Quot schemes, providing explicit descriptions of their cones and proving they are Mori dream spaces.

Findings

Quot schemes are Mori dream spaces.

Explicit descriptions of nef, movable, and effective cones.

Determinant morphism is a Mori dream morphism.

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.