Let $X \subset \mathbb{P}^n$ be a hypersurface with $n \ge 3$. Let $x \in X$ be a closed point. Consider the map given by projection from $x$: $$\phi: X \dashrightarrow \mathbb{P}^{n-1}$$ Suppose that $\phi$ is not dominant. Does this imply that $X$ is singular at the point $x$? The reason for asking this question is because heuristically it seems that in this case the fibers are isomorphic to $\mathbb{P}^1$ passing through $x$, which would mean that near $x$, $X$ looks like a cone over a hypersurface in $\mathbb{P}^{n-1}$.
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2$\begingroup$ If this map is not dominant, then $X$ must be a cone with vertex at $x$. $\endgroup$– Lazzaro CampeottiCommented Dec 6 at 16:11
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2$\begingroup$ How about a hyperplane through $x$? (That said, this is the only counterexample.) $\endgroup$– Daniel LittCommented Dec 6 at 16:12
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$\begingroup$ @DanielLitt: Thanks! Thats a very nice counterexample :) However, I am mainly interested in hypersurfaces of degrees 2 and higher. $\endgroup$– Naga VenkataCommented Dec 6 at 16:16
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