# When is the projection from a point on the variety smooth?

Let $$X\subset\mathbb{P}^n$$ be a smooth irreducible variety (over $$\mathbb{C}$$) and $$p\in\mathbb{P}^n$$ a point. Let $$\pi:\mathbb{P}^n\setminus\{p\}\to\mathbb{P}^{n-1}$$ be the linear projection with center $$p$$ and denote by $$Y$$ the Zariski closure of $$\pi(X\setminus\{p\})$$. I would like to know:

When is $$Y$$ smooth and the restriction $$X\setminus\{p\}\to\pi(X\setminus\{p\})$$ of $$\pi$$ an isomorphism?

If $$p\not\in X$$, then I think this is equivalent to $$p$$ not lying on the secant variety of $$X$$. But I am mainly interested in the case when $$p\in X$$. In this case we can rephrase the question as:

When is $$Y$$ the blow-up of $$X$$ at $$p$$?

• If you require $\pi$ being generically one-to-one and the projection center on the variety, then $X$ has degree 2. I think it only happens when $X$ is a smooth quadric hypersurface. – AG learner Aug 4 at 18:31
• No, I don't think so. For example if $X$ is the rational normal curve of degree $n$, then $Y$ is the rational normal curve of degree $n-1$ and the restriction of $\pi$ is an isomorphism onto its image. Maybe there was an ambiguity in the question which I edited now. – Hans Aug 5 at 7:18
• Sorry, you’re right. I was thinking about codimension one case. – AG learner Aug 5 at 15:12

Lemma: Let S be a surface in $$\mathbb{P}^N$$ and $$p \notin S$$ (respectively $$p \in S$$). Let $$f: S \rightarrow \mathbb{P}^{N-1}$$ (respectivly $$f: \hat{S} \rightarrow \mathbb{P}^{N-1}$$) be the restriction of the projection away from $$p$$. Then $$f$$ is an embedding $$\iff$$ there is no line through $$p$$ meeting $$S$$ in at least $$2$$ (respectively at least $$3$$) points, counted with multiplicitly.
A nice corollary of this statement (given immediately after in Beauville's book) is that every smooth projective surface is isomorphic to a surface in $$\mathbb{CP}^5$$.
• Every smooth projective surface is isomorphic to a surface in $\mathbb{P}^5$? – red_trumpet Aug 5 at 8:49