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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!

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$A_{10}$ for a plane quintic doesn't seem all that surprising: quintic curves move in a family of dimension $6\cdot 7/2 - 3^2 = 12$, and for each $m$ the curves with an $A_m$ singularity should form a subfamily of codimension $m$, so one expects that $m$ can get as large as $12$. Indeed the paper <> exhibits on page 20 the quintic $x^3-2(xz)^2+zx^4 + 4xz + 4 − 4x^3 = 0$ and asserts it has an $A_{12}$ singularity at infinity. In general one expects for the same reason that plane curves of degree $d$ could have $A_m$ singularities for $m$ as large as $(d+1)(d+2)/2-9$. – Noam D. Elkies Jan 12 '12 at 22:33
I suppose that here by codimension you mean the codimension of the equisingular stratum in the base of the versal deformation of the isolated singularity. In that case, let $T$ be a singularity with codimension d < 40. The quintic surfaces moves in a family of dimension 40. Does your argument implies that the quintic surfaces could have a $T$ singularity? In singularities are classified by their "jacobian algebra" so I was thinking to find generators of this C-algebra on a given degree. Unfortunately, I dont know about that this technology! – eventually Jan 13 '12 at 1:26
All I'm suggesting is that - barring evidence or heuristics to the contrary - it wouldn't be too surprising if a quintic surface actually had a $T$ singularity. As for actual proofs, I don't know this "technology" either, and will leave it to those who do to post actual answers rather than comments. – Noam D. Elkies Jan 13 '12 at 16:48
...though for quartic surfaces the Torelli theorem for K3 surfaces gives very precise information about the configurations of ADE singularities that can arise; likewise for sextic plane curves, because the double cover of ${\bf P}^2$ branched along a sextic with ADE singularities is birationally K3. – Noam D. Elkies Jan 13 '12 at 17:14

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.

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Concerning plane curves with fixed degree $d$ and $A_m$ singularities such that $m>d$, I'd also mention the following tiny note by Gusein-Zade and Nekhoroshev: arXiv:math/9906147 – Anton Fonarev Jan 14 '12 at 5:10

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.

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