On birational geometry of the space of parametrized rational curves in Grassmannians

TL;DR

This paper explores the birational geometry of Quot schemes of trivial bundles on projective lines, establishing their minimal models, Mori dream space status, and log Fano properties, thus advancing understanding of their geometric structure.

Contribution

It constructs small Q-factorial modifications of Quot schemes and classifies all models in their minimal model program, revealing their Mori dream space and log Fano characteristics.

Findings

Quot schemes are Mori dream spaces.

All models in the minimal model program are classified.

Quot schemes are log Fano varieties.

Abstract

In this paper, we study the birational geometry of the Quot schemes of trivial bundles on $\mathbb{P}^1$ by constructing small $\mathbb{Q}$-factorial modifications of the Quot schemes as suitable moduli spaces. We determine all the models which appear in the minimal model program on the Quot schemes. As a corollary, we show that the Quot schemes are Mori dream spaces and log Fano.