# ag.algebraic geometry – Normal bundle to Veronese varieties \$v_d(mathbb{P}^n)\$ into \$mathbb{P}(H^0(mathcal{O}_{mathbb{P}^n}(d)))\$

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.