Map of Grassmannians associated with a Veronese embedding

Question

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

Answer 1

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 $$ n\epsilon_1 + ((n-1)\epsilon_1 + \epsilon_2) + \dots (\epsilon_1 + (n-1)\epsilon_2) + n\epsilon_2 = \binom{n+1}{2}\omega_2 $$ (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.