Let $Gr(1,n)$ be the Grassmannian of lines in $\mathbb{P}^n$, and $f:\mathbb{P}^1\rightarrow Gr(1,n)$ a morphism of degree two. The pull-back $f^{\*}S$ of the tautological bundle $S$ on $Gr(1,n)$ splits as $f^{\*}S = \mathcal{O}\_{\mathbb{P}^1}(a)\oplus \mathcal{O}\_{\mathbb{P}^1}(b)$. If $f$ is general in the moduli space of degree two morphisms $\mathbb{P}^1\rightarrow Gr(1,n)$ then $a = b = -1$. However, for some morphisms $f$ we may have $a = 0, b = -2$.


Is there any geometric characterization of the morphisms $f:\mathbb{P}^1\rightarrow Gr(1,n)$ yielding $a = 0,b = -2$?



Jumping conics in Grassmannians
These conics are exactly conics contained in some
$$
\mathbb{P}^{n-1} \subset Gr(1,n),
$$
that parameterizes all lines in $\mathbb{P}^n$ passing through a fixed point.


