Let $\phi:(X,x)\to (T,t)$ be a morphism of germs of algebraic schemes over $\mathbb{C}$.
How can we always (without "quasi-projective") have an open embedding of the germ $(X,x)$ into a closed subscheme $X'\subset T\times \mathbb{P}^n$ for some $n$? It suffices to consider $\phi':U\to T$ where $U$ is an affine open neighborhood of $x$ in $X$?