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Let $X$ be a finite type scheme over a field $k$. For a point $x\in X$ with the local ring $\mathcal{O}_x$ we define the tangent cone at $x$ as the spectrum of the ring: $$\mathrm{gr}_{\mathfrak{m}_x}(\mathcal{O}_x)$$ It is well known fact that the krull dimension of $\mathcal{O}_x$ and of the tangent cone are the same. There are also many examples concerning plane curves which show that the tangent cone says something about local behaviour of $X$ at $x$.

$Question:$ How much information concerning local behaviour of $X$ at $x$ does tangent cone carry?

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  • $\begingroup$ I may be missing something, but in the case of plane curves the tangent cone says almost nothing about the local behavior. $\endgroup$ Jan 30, 2014 at 21:08
  • $\begingroup$ One obvious thing you can say is this: A variety is smooth at $x$ iff the tangent cone is the symmetric algebra on the zariski cotangent space. $\endgroup$ Jan 31, 2014 at 2:21
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    $\begingroup$ The plane curves $X$ defined by $y(y-x^n)$, $n\geq 2$ all have the same tangent cone. I think a reasonable answer might be "first order information". $\endgroup$ Feb 1, 2014 at 18:53

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