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?