canonical bundle of Veronese embedding

Question

Suppose we are given a complete intersection $X$ of codimension $r=n-d$ in $\mathbb{P}^n$ where the degrees of the hypersurfaces are $d_i$ and $d$ is the dimension of $X$. Then the canonical bundle $\omega_X=\mathcal{O}_X(\sum d_i-n-1)$. Now suppose we embedd $X$ via a Veronese embdding in some $\mathbb{P}^N$ for some $N>>0$ so that one has inclusion $i:X\rightarrow \mathbb{P}^N$. By Hartshorne Proposition 8.20 one can calculate $\omega_X$ is this case but what is the normal bundle $\mathcal{N}_{X/\mathbb{P}^N}$ in this case?

Answer 1

Suppose $M$ subset of $\Bbb R^m$ is a closed embedded submanifold. if $M$ admits a global defining function, show that is normal bundle is trivial. Conversely, if $M$ has trivial normal bundle, show that there is a nighborhood $U$ of $M$ in $\Bbb R^n$ and a submersion $K$ from $U$ to $\Bbb R^k$ such that $M=K^{-1}(U)$.