Qfyilyi

I'm quite sure this should be classically known, however I am not an expert of the topic and I was unable to find a precise reference in the huge literature concerning Veronese embeddings and Grassmannians. Any pointer to the relevant literature will be appreciated.

Let $V$ be a finite-dimensional vector space (over $mathbb{C}$, say) and let $v_n colon mathbb{P}(V) longrightarrow mathbb{P}(S^n V)$ be the usual $n$th Veronese embedding.

If $ell$ is a line in $mathbb{P}(V)$, then $v_n(ell)$ is a rational normal curve of degree $n$, that will be contained in precisely one $n$-plane $Pi_{ell} subset mathbb{P}(S^n V)$. Then we can define a morphism of projective Grassmannians $$psi_n colon mathbb{G}(1, , mathbb{P}(V)) longrightarrow mathbb{G}(n, , mathbb{P}(S^nV)),$$
given by $psi_n(ell) :=Pi_{ell}$.

Question. Is $psi_n$ an embedding?

I don't know a reference, but here is a simple argument. Note that $G(2,V)$ (let me use linear notation) is a homogeneous space for $GL(V)$:
$$
G(2,V) = GL(V)/P_2,
$$
where $P_2$ is a parabolic. If $e_1,dots,e_N$ is the basis of $V$, we can take $P_2$ to be the stabilizer of the point
$$
p_1 := [e_1 wedge e_2] in mathbb{P}(wedge^2V).
$$
Note that $e_1 wedge e_2$ is the highest weight vector with weight
$$
epsilon_1 + epsilon_2 = omega_2
$$
(the second fundamental weight of $GL(V)$).

The map $psi_n$ is $GL(V)$-equivariant, and takes $[e_1 wedge e_2]$ to
$$
p_n := [(e_1^n) wedge (e_1^{n-1}e_2) wedge dots wedge (e_1e_2^{n-1}) wedge (e_2^n)].
$$
It is easy to check that this is a highest vector with weight
$$
nepsilon_1 + ((n-1)epsilon_1 + epsilon_2) + dots (epsilon_1 + (n-1)epsilon_2) + nepsilon_2 = binom{n+1}{2}omega_2
$$
(it corresponds to an irreducible subrepresentation $V_{binom{n+1}{2}omega_2} subset wedge^{n+1}(S^nV)$), and its stabilizer is the same parabolic subgroup $P_2$. It follows that $psi_n$ is an isomorphism onto the orbit of the point $p_n$, in particular it is an embedding.