I was searching for a response on the internet but I was not able to find out an explicit answer.

It is known that if $mathbb{P}^n subset mathbb{P}^N$ is embedded linearly then the normal bundle $N_{mathbb{P}^n/mathbb{P}^N}cong mathcal{O}_{mathbb{P}^n}(1)^{oplus (N-n)}$.

This can be proved for example via Koszul complex.

My question now is the following: if we embed $mathbb{P}^n$ into $mathbb{P}^N$ with higher degree, for example with the Veronese embedding

$$v_d:mathbb{P}^n rightarrow mathbb{P}^N:=mathbb{P}(H^0(mathcal{O}_{mathbb{P}^n}(d)))$$

what is it the normal bundle $N_{v_d(mathbb{P}^n)/ mathbb{P}^N}$? It is possible that could be $mathcal{O}_{mathbb{P}^n}(d)^{oplus(N-n)}$?

I was trying some Koszul approach like in the linear case but for $d>1$ I’m not able anymore to control the free resolution of the Veronese varieties.

Thanks in advance.