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?
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 ?