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 $\mathcal{O}\_{v\_d(\mathbb{P}^n)}(d)^{\oplus(N-n)}$?


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.



Normal bundle to Veronese varieties $v_d(\mathbb{P}^n)$ into $\mathbb{P}(H^0(\mathcal{O}_{\mathbb{P}^n}(d)))$
The normal bundle $N$ of the Veronese embedding $\mathbb{P}(V) \to \mathbb{P}(S^dV)$ can be described by the exact sequence
$$
0 \to V \otimes \mathcal{O}(1) \to S^dV \otimes \mathcal{O}(d) \to N \to 0,
$$
where the first arrow is the unique nonzero $\mathrm{GL}(V)$-equivariant morphism.


Alternatively, one can describe the normal bundle as an iterated extension of symmetric powers of the tangent bundle $T$ of $\mathbb{P}(V)$ --- there is a filtration on $N$ with associated graded of the form
$$
\mathrm{gr}\_\bullet(N) = \bigoplus\_{i=2}^d S^i T.
$$


Finally, let me give a couple of explicit examples. If $\dim(V) = 2$ one has
$$
N\_{\mathbb{P}(V)/\mathbb{P}(S^dV)} \cong S^{d-2}V \otimes \mathcal{O}(d+2),
$$
and if $d = 2$ one has
$$
N\_{\mathbb{P}(V)/\mathbb{P}(S^dV)} \cong S^2T.
$$


