On the loci of morphisms from $\mathbb{P}^1$ to $G(r,n)$ with fixed splitting type of the restricted universal sub-bundle or quotient bundle

TL;DR

This paper investigates the geometric loci of morphisms from the projective line to Grassmannians with fixed splitting types of universal bundles, revealing non-emptiness, transversality, and implications for irreducibility.

Contribution

It establishes non-emptiness and generic transversality of loci with fixed splitting types, and shows the locus of morphisms with a fixed tangent bundle splitting type can be reducible.

Findings

Loci with fixed splitting types are non-empty.

Loci are generically transverse.

The locus with fixed tangent bundle splitting type can be reducible.

Abstract

Let $n\geq 4$, $2 \leq r \leq n-2$ and $e \geq 1$. We show that the intersection of the locus of degree $e$ morphisms from $\mathbb{P}^1$ to $G(r,n)$ with the restricted universal sub-bundles having a given splitting type and the locus of degree $e$ morphisms with the restricted universal quotient-bundle having a given splitting type is non-empty and generically transverse. As a consequence, we get that the locus of degree $e$ morphisms from $\mathbb{P}^1$ to $G(r,n)$ with the restricted tangent bundle having a given splitting type need not always be irreducible.