Let $X$ be an irreducible subvariety of dimension $d$ in $\mathbb P^n$. Can we find a linear projection $\pi:\mathbb P^n \dashrightarrow \mathbb P^{d+1}$ such that $\pi: X \to \pi(X)$ is a finite and regular birational map?



Existence of a linear projection
By induction it is enough to check that if $n \ge d + 2$ there is a linear projection $\mathbb{P}^n \dashrightarrow \mathbb{P}^{n-1}$ that has all required properties. For this consider the subvariety
$$
S = \{(p,\xi) \in \mathbb{P}^n \times X^{[2]} \mid 
p \in \langle \xi \rangle \}.
$$
Here $X^{[2]}$ is the Hilbert square of $X$, so that $\xi$ is a subscheme of $X$ of length $2$, and $\langle \xi \rangle$ is the unique line in $\mathbb{P}^n$ containing $\xi$. Note that
$$
\dim(S) \le \max(2d + 1, d + n - 1) = d + n - 1
$$
(meaning that every irreducible component of $S$ enjoys this bound). Indeed, the open subset in $S$ that corresponds to $\xi = \{x\_1,x\_2\}$ with $x\_1 \ne x\_2$ has dimension $2d - 1$, and the closed subset where $\xi$ is a tangent vector has dimension at most $d + n - 1$ ($d$ parameters for a point and at most $n - 1$ for the tangent direction).


Now consider the morphism
$$
S \to \mathbb{P}^n,
\qquad
(p,\xi) \mapsto p.
$$
By the dimension bound there is a point $p \in \mathbb{P}^n \setminus X$ where the fiber has dimension at most
$$
d + n - 1 - n = d - 1.
$$
The linear projection from such a point works, because at most $(d - 1)$-dimensional set of points in the image of $X$ have more than one point in the preimage.


