On the loci of morphisms from [formula omitted] to G(r,n) with fixed splitting type of the restricted universal sub-bundle or quotient bundle

Abstract

Let n≥4, 2≤r≤n−2 and e≥1. We show that the intersection of the locus of degree e morphisms from P1 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 along at least one component of the intersection. As a consequence, we get that the locus of degree e morphisms from P1 to G(r,n) with the restricted tangent bundle having a given splitting type need not always be irreducible.