embedding complex submanifolds in projective space

Question

Suppose you have a projective manifold M and a very ample bundle L and so a transvese holomorphic section s. The zero set will be a complex submaifold S_M. Can we have a embedding of the the projective manifold M in some projective space such that image of S_M will not be contained in a hyperplane.

Answer 1

If the embedding is induced by $L$, then no by definition of how the embedding is defined (the zero locus of your non-zero sections correspond to hyperplane sections).

If the embedding is not induced by $L$, then sure. Take $M=\mathbb{P}^1$ and $L=\mathcal{O}(2)$, consider the embedding from $M$ to itself. A hyperplane is just a point, but the zero locus of a generic non-zero element of $L$ consists of two points.