How many types of surface singularities multiplicity two exist?

Question

All varieties are over $\mathbb{C}$.

Let $S$ be a reduced algebraic surface in $\mathbb{P}^3$ with a singular point $p$ of multiplicity two. The question is local so we reduce to $S \subset \mathbb{A}^3$ and $p=0$. Then $S$ is given as the zero set of a polynomial $f = f_2 + f_3 \ldots$ where $f_i$ is homogeneous of degree $i$ and $f_2$ is nonzero (Hartshorne, exercise I.5.3).

The question in the title is vague, so let me split it into parts:

• If the singularity $p$ is isolated, is it necessarily an ordinary quadratic one? (i.e. is it resolved by a single blowup operation, resulting in a rational -2 curve?) If not, what are examples of other possibilities?

• Is it true that $p$ is isolated if and only if its two tangent directions (the linear factors of $f_2$, see again Hartshorne exercise I.5.3) are different?

These two questions are based on intuition and not much more, so i hope they are not totally ridiculous. When answering, any elaboration on the concept of multiplicity and tangent directions is highly appreciated! Thanks!

Edit: After learning about Du Val singularities, i realize the first question is pretty stupid.. But then again, the question could be replaced by "Are the Du Val singularities the only surface singularities of multiplicity two?"

Answer 1

For a surface, the multiplicity of a singular point is the first invariant to look for. Suppose it is a double point. In that case, you want to look for the tangent cone. If the tangent cone is $xy$ then you have an $A_k$ singularity (basically of the form $x^2+y^2+z^k$). If the tangent cone has the form $x^2$. Then, the double point singularity has the form $$ x^2+g(y,z) $$ Now, you can use other invariants such a the Milnor number, the modality, etc... A good place to star is " Singularities of Differentiable Maps: Volume 1: The Classification of Critical Points Caustics, Wave Fronts by V.I. Arnold, S.M. Gusein-Zade, A.N. Varchenko". Go to the index, and look for classification of singularities.