# Reference request: Projection from a general point on a variety

Let $$X$$ be a smooth projective variety in $$\mathbb{P}^n$$, over the field of complex numbers or an algebraically closed field of characteristic $$0$$.

EDIT (After Stefano's response): Assume $$\dim X < n-1$$, $$\deg X > 1$$ and $$X$$ is nondegenerate.

In particular, $$X$$ is not a hypersurface. Let $$P\in X$$ be a general point, fix a general hyperplane in $$\mathbb{P}^n$$ and consider the map $$\varphi_P:X\setminus \{P\}\to \mathbb{P}^{n-1}$$ obtained by "projecting from $$P$$". Let $$Y$$ be the closure of the image $$\varphi_P(X)$$ in $$\mathbb{P}^{n-1}$$.

(1) For general $$P$$, is the map $$\varphi_P:X\setminus\{P\}\to Y$$ going to be quasi-finite?,

(2) For general $$P$$, is the map $$\varphi_P:X\setminus\{P\}\to Y$$ going to be birational?

I am guessing that these are true. I could not find a reference for these two statements in standard texts like Harris, Hartshorne or Shararevich . Can someone please point out a reference for these. If someone could sketch a proof or explain a counterexample then that would be more helpful. Thanks.

One condition that you need is that $$X$$ is non-degenerate, i.e. it is not contained in any hyperplane. Indeed you can consider a plane cubic $$E$$ as a curve in $$\mathbb{P}^3$$. Then, the projection will be $$2:1$$, since every line will meet two residual points. Thus, the map is not birational.
Similarly, you can consider a quadric surface $$Q \subset \mathbb{P}^3$$ as embedded in $$\mathbb{P}^4$$. Then, through $$P$$ there are exactly two rulings ($$Q$$ is isomorphic to $$\mathbb{P}^1 \times \mathbb{P}^1$$), and the map contracts each ruling to a point. Thus, the map is not quasi-finite.