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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 $a = 0,b = -2$?

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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.

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    $\begingroup$ Thank you for the answer. Can you explain why for instace for conics in $Gr(1,3)$ spanning a plane contained in $Gr(1,3)$ we have $a = 0,b= -2$? $\endgroup$
    – Puzzled
    Commented Apr 3, 2021 at 20:40
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    $\begingroup$ Because the restriction of $S$ to the plane has a nonzero global section (that corresponds to the fixed point of lines parameterized by this plane). $\endgroup$
    – Sasha
    Commented Apr 3, 2021 at 20:45

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