Let $V=\mathbb{C}^2$. $PGL_2(\mathbb{C})$ acts naturally on $V$. This action induces action of $PGL_2$ on $sym^n(V)$.
Let $\iota_n: \mathbb{P}(V)=\mathbb{P}^1\to \mathbb{P}^n=\mathbb{P}(sym^n(V))$ be the natural Veronese embedding where line passing through vector $v\in V $ mapped to line passing through vector $v^n\in sym^n(V)$.
It is evident that $PGL_2(\mathbb{C})$ action on $\mathbb{P}(sym^n(V))$ will map $[v^n]$ to $[(g\cdot v)^n]$ hence preserves the image
Im$(\iota_n):=C_n:=\{[v^n] : v\in V\}\subset \mathbb{P}(sym^n(V))$, the rational normal curve of degree n.
- Why any automorphism of $\mathbb{P}^n$ fixing $C_n$ point wise has to be identity?
- Why the group G of automorphisms of $\mathbb{P}^n$ which preserve $C_n$ is precisely PGL_2?