Describe $\mathcal{N}_{G(\mathbb{P}^1,\mathbb{P}^k)\mid G(\mathbb{P}^1,\mathbb{P}^n)}$ [from MSE]

Question

Note: This question came from MSE, but since I've received some useful observations I posted it here. Post on MSE

Consider $1 \leq k < n$ positive integers, and denote by $G(\mathbb{P}^k,\mathbb{P}^n)$ the Grassmannian of $\mathbb{P}^k$'s in $\mathbb{P}^n$. It is well known $G(\mathbb{P}^k,\mathbb{P}^n)$ admits a structure of projective variety in $\mathbb{P}^N$, where $N=\binom{n+k}{2}-1$.

It is moreover known that, denoting by $\mathcal{N}_{\mathbb{P}^k\mid \mathbb{P}^n}$ the normal bundle of $\mathbb{P}^k$ in $\mathbb{P}^n$, it holds the following

$$\mathcal{N}_{\mathbb{P}^k\mid \mathbb{P}^n} =\mathcal{O}_{\mathbb{P}^k}(1)^{\oplus n-k}.$$

I was wondering if there exists a similar result if we consider a Grassmannian instead of the projective space, that is

$$ \text{ } \mathcal{N}_{G(\mathbb{P}^1,\mathbb{P}^k)\mid G(\mathbb{P}^1,\mathbb{P}^n)}=\mathcal{O}_{G(\mathbb{P}^1,\mathbb{P}^k)}(1)^{2(n-k)} \text{ }??$$

I've put the double question mark to stress this may be completely nonsense, but it's my guess (nothing rigorous).

Comments:

• The $2(n-k)$ came from a comment on MSE and it is based on a simple computation of dimension of Grassmannian.

• In order to talk about $\mathcal{O}_{G(\mathbb{P}^1,\mathbb{P}^k)}$ I need a projective embedding, but there's the natural Plucker embedding we can consider.

I was wondering if there exists such a similar description. I've thought about it for a while but didn't manage to find anything.

Answer 1

(A more general answer valid not only for the Grassmannian of lines).

Denote by $U$ the rank $k$ tautological bundle on $Gr(k,n)$, using the convention that $det(U^{\vee})= \mathcal{O}_{Gr(k,n)}(1)$ (in the Pluecker embedding). Then the zero locus of a general global section of $U^{\vee}$ is naturally identified with $Gr(k, n-1)$. In particular $$\mathcal{N}_{Gr(k, n-1)| Gr(k,n)} \cong U^{\vee}_{Gr(k,n)}|_{Gr(k, n-1)} \cong U^{\vee}_{Gr(k, n-1)}.$$

(notice however that for the quotient bundle one has $Q_{Gr(k,n)}|_{Gr(k, n-1)} \cong Q_{Gr(k, n-1)} \oplus \mathcal{O}$).

You can iterate this construction taking as many copies of $U^{\vee}$ as you wish to get to the desired Grassmannian.

(Notice moreover that in the case $Gr(1, n+1)= \mathbb{P}^n$ this normal bundles coincides with the obvious hyperplane section).