The effective cone of the space of parametrized rational curves in a Grassmannian

TL;DR

This paper explicitly determines the effective cone of the Quot scheme parametrizing rank r, degree d quotient sheaves of the trivial bundle on P^1, by constructing spanning divisors and expressing their classes.

Contribution

It provides an explicit description of the effective cone of the Quot scheme for parametrized rational curves in a Grassmannian, including construction and class expressions of key divisors.

Findings

Constructed two effective divisors spanning the cone

Expressed divisor classes in the Picard group

Explicit description of the effective cone

Abstract

We determine the effective cone of the Quot scheme parametrizing all rank r, degree d quotient sheaves of the trivial bundle of rank n on P^1. More specifically, we explicitly construct two effective divisors which span the effective cone, and we also express their classes in the Picard group in terms of a known basis.