Let us consider the following specific problem for blowing-ups.
Let $n$ be a large positive integer. Let $X\subset \mathbb P^n$ be a smooth sub variety of codimension $>1$. Denote $Y$ the blowing-up of $\mathbb P^n$ along $X$. I would like to inject $Y$ into $\mathbb P^{n+1}$ (thus as a hyper surface). Is it something possible ? If not, what could be the minimal $N$ (depending on $n$ and $X$ of course) such that $Y$ can be embedded into $\mathbb P^N$ ?
Ideas, examples and references are welcome ! Thanks in advance.
p.s. By saying that $n$ is large, I mean that I wish to have results for $n$ as large as possible. But I do not even know if the value of $n$ is really important in this question. One may wish to restrict to complex projective varieties to fix ideas.