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Say we have a plane curve $\mathcal{C} = V(f(x,y)) \subset \mathbb{A}^2_{\mathbb{C}}$. The partial derivatives tell us about the singularities: if they all vanish at a point $p \in\mathcal{C}$ then the curve is singular at this point. My question is: do the partials tell us what kind of singularity there is? That is, would we be able to detect a cusp, node, etc. just by looking at the partials?

More generally, if we have some space curve, $\mathcal{C} \subset \mathbb{A}^n_{\mathbb{C}}$, the minors of the Jacobian cut out the singular locus. Can they tell us what kind of singularity we have?

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No you can't detect the singularity type with the Jacobian. Did you try with the simplest example of $y^2-x^3$ and $y^2-x^2(x+1)$ ?

The Jacobian only tells you about the dimension of the tangent spaces. The Hessian will say more (it gives information on the tangent cone). How happens with the above example ?

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    @unit3000-21, the singularity (cusp) in the first case is in some sense worse than the second one (ordinary double point). This is as for one-variable function: the first has vanish order $\ge 3$ and the second has order $=2$. This difference is detected by the Hessian.2012-10-11