How can we find a surface with a given singularity?  I was surprised the first time I learned that a quintic plane curve can have an $A_{10}$ singularity i.e $x^2+y^{10}$.  I am wondering if there is something about that phenomenon: Given a singularity on its normal form, and a fixed degree $d$. Is there a standard way to find an hypersurface of degree $d$ on $\mathbb{P}^n$ with the given singularity? Is there a lower bound for $d$ in terms of the singularity invariants?
For example: Given the singularity $x^2+y^3+z^{13}$ and $d=5$, the surface
$$ 
(x+z^3)^2+(y-z^2)^3+x^3y^2+x^5
$$
has that singularity at $(0,0,0)$.  How can I define a surface for another singularity e.g. $x^2+y^4+z^{22}$ ?
I am mostly thinking in surfaces and plane curves.  Thanks for any hint or suggestion!
 A: For plane curves, general sufficient conditions have been given by Shustin (Trans. AMS 356, 2004, 953–985) although for particular singularity types (such as A-singularities) sharper results are known (see J. Alg. 302, 2006, 37-54). For one single $A_m$ singularity, I think the best sufficient condition is due to Lossen, via explicit equations (EDIT: Comm. Algebra 27, 1999, 3263–3282). In general it is not enough that the linear system of plane curves of degree $d$ has dimension at least equal to the codimension of the singularity type (except for the case of $m$ nodes, when this is necessary and sufficient).
In higher dimension, less is known, but again I'd suggest to look at Shustin-Westenberger, J. London Math. Soc. 70, 609–624.
A: I'm asumming you assume ground field $\mathbb{C}$. I actually wondered about the same thing a while ago, in the case of surfaces. I find hard to think in particular embeddings a priori and then the singularities on it but I'd rather think first on the singularities and then think where they can be embedded in. You speak of $A_n$ which in the case of surfaces is a Du Val singularity and it is given by certain equations anallytically which you can find in Reid's notes. However if you start combining several singularities in the same surface and for instance fix the degree, it is intuitive that you cannot glue the different analytic patches together. It is therefore 'easier' to start with a surface with given singularities, degree, Picard number... and check whether it exists or not.
A classification of log Del Pezzo surfaces of index $\leq 2$ was done by Alexeev and Nikulin. I think more general cases are unknown. Higher dimensional cases are probably even more complicated. This classification has been used for instance, to classify certain 3-folds with $T$-singularities (see Hacking-Prokhorov)
This is not a great answer, but maybe it helps to point you on references to read upon. A particular example may be easy, but a general picture I am afraid that requires a long road for which I have not read the map completely.
