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Jul 15, 2016 at 10:49 vote accept Irfan Kadikoylu
Jul 8, 2016 at 16:40 answer added abx timeline score: 2
Jul 8, 2016 at 12:18 comment added Lazzaro Campeotti Just a comment: If $X$ is not a cone, the lines in $X$ through $p$ fill up at most a codimension-1 subset of $X$, hence $n$ is at most $r-3$. But for example there are smooth cubic threefolds in $\mathbf P^4$ with Eckardt points, i.e. points where the tangent hyperplane section is a cone (over a curve, ie $n=1$). So $n=r-3$ is not sufficient in general.
Jul 8, 2016 at 11:34 history asked Irfan Kadikoylu CC BY-SA 3.0