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

Abstract

Let $n\ge 4$, $2\le r\le n-2$ and $e\ge 1$. 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.

22 pages