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X in P^n is a cone iff there exists point P on X such that multiplicity of X at point P is 2. The hint for this problem is that a tangent space of hypersurface X=V(F) at P is union of lines L which are tangent in point P in a sense that restriction of F to L has multiplicative root at P.

Im not sure how to use this hint to solve the problem, and I would appriciate any help.

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  • $\begingroup$ First, there are cones which are non-singular, like a linear space. So, to get what you want, usually one assumes that the variety is not contained in a hyperplane. But, your title is about quadrics (which is not in the main body). In this case, if it has a singular point, just project from this point to $n-1$ space and use Bezout's theorem to see that the quadric is in fact the cone over the image under the projection with your singular point as the vertex. $\endgroup$ – Mohan Feb 13 '17 at 1:18

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