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Given an isolated singularity $p$ in a hypersurface $Y$ of dimension $n$ (let say a surface in $\mathbb{P}^3$). I can intersect $Y$ with a hyperplane $H$ passing through $p$ such that it induces a singularity in a lower dimension. For example, if we start with a surface $Y$, we obtain a plane singularity on $H$.

I want to use the singularities in the hyperplanes for classifying(?) the original singularity in the hypersurface. Using homeomorphism for defining an equivalence relation. There is a finite number of singularities types that I can obtained in the hyperplanes, and one of those types will be "generic".

I want to believe that those hyperplane singularities + some "vicinity" data is enough for characterizing the original singularity. However, I cannot find related work, or theorems to start with. I appreciate any help or references.

Thanks

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This is a good strategy and it does work for many types of singularities. In fact, many times you don't even need "vicinity" data if your singularity is isolated. (In other words, being isolated is the "vicinity" data). If you do not assume that the singularities are isolated, then you need to assume something about $X\setminus H$.

Typical theorem

Let $X\subseteq \mathbb P^N$ be a quasi-projective variety and let $H\subseteq X$ be a hyperplane section and assume that $H$ only has singularities of type $\mathfrak T$. Then $X$ only has singularities of type $\mathfrak T$ along $H$. In particular, if in addition $X\setminus H$ only has singularities of type $\mathfrak T$, then so does $X$.

Obviously, whether or not the above theorem is in fact true depends on your choice of $\mathfrak T$. It is known in many cases and conjectured in others.

If singularities of type $\mathfrak T$ satisfy the general inverse of the above, that is if the

General hyperplane section theorem

Let $X\subseteq \mathbb P^N$ be a quasi-projective variety and let $H\subseteq X$ be a general hyperplane section and assume that $X$ only has singularities of type $\mathfrak T$. Then $H$ only has singularities of type $\mathfrak T$.

holds, then the two theorems together imply that small deformations of varieties with singularities of type $\mathfrak T$ still have singularities of type $\mathfrak T$.

In other words, a possible way to find the statement you are looking for is to look for papers that claim that certain singularity types are invariant under small deformations.

Here is a list of singularities for which both of these theorems hold:

  • smooth (Sketch of proof): being smooth means that the local ring is regular. If a ring mod a regular element is regular, then so is the ring.

  • Cohen-Macaulay : basically by the definition of CM

  • Gorenstein : Gor = CM + $\omega$ is a line bundle. CM follows from above and line bundle from the fact that if the restriction of a coherent sheaf to a hypersurface is a line bundle, then so is the original sheaf.

  • rational : This is a result of Elkik (Inventiones, cca. 1978)

  • klt,dlt,lc,etc : this is essentially inversion of adjunction

  • Du Bois : this is a result of Kovács-Schwede (available on arXiv.org)

Remark

In fact, there is a tendency toward $X$ having better singularities than $H$ does. Namely it is sometimes true that (for example)

If $H$ only has singularities of type $\mathfrak T$ and $X\setminus H$ is smooth, then $X$ only has singularities of type $\mathfrak T^+$ for some class of singularities that is milder than $\mathfrak T$.

An example for this is Karl Schwede's theorem (Thm.5.1 in A simple characterization of Du Bois singularities, Compositio Math. 143 (2007) 813–828) that says exactly this with $\mathfrak T$ being "Du Bois" and $\mathfrak T^+$ being "rational.

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I wanted to add a more elementary example of the phenomenon in the last remark. The statement there is also true if we take $\mathfrak T$ to be "reduced" and $\mathfrak T^+$ to be "normal." –  Dustin Cartwright Oct 23 '11 at 20:58
    
That's right. And more generally it works with $\mathfrak T$ being $S_n$ and/or $R_k$ and $\mathfrak T^+$ being $S_{n+1}$ and/or $R_{k+1}$. –  Sándor Kovács Oct 23 '11 at 21:30
    
Thanks a lot for the answers and references. I was also wondering in the line of Arnolds's classification of singularities. Indeed, the Arnold's list, plus C.T.C Wall's work provide a complete classification up to modality ~3, and multiplcity ~6, but higher multiplicity singularities classification seems incomplete. However, I acknowledge those singularities are uglier than usual, and not that often exposed. Any case, thanks. –  pmath Oct 29 '11 at 0:57
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V. I. Arnol'd's list of isolated boundary singularities $B_k,C_k,F_4$ has been extended by V. I. Matov to include also unimodal $F_{1,0},...$, e.t.c. singularities. This covers a long list which answers to the problem of hyperplane section singularities up to some reasonable codimension. Notice that in your notation, even if $Y$ is smooth at $p$, the intersection with the hyperplane $H$ might not be transversal. This leads to the ordinary $A_k,D_k,E...$ series of singularities of $H\cap Y$. The relation of the singularities of the pair $Y,Y\cap H$ has been studied by I. Scherback (Legendre, Lagrange transforms and duality). A good reference for all these is, as usual, Arnol'd-Gusein-Zade: Singularities of Differentiable Maps, Vol. I & Vol. II and references therein.

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  1. Not clear why do you want only the hyperplane sections (and not all the smooth hypersurface sections of the initial singularity). For example, the plane curve singularity $\{y^2=x^k\}$, $k>2$ give the same sections if one intersects by lines only.

  2. If you consider the intersection by all the smooth hypersurface germs, then you have more chances to restore the original (embedded toplogical) singularity type. (Especially if you are interested only in some simplest-type singularities.) But in general just the collection of all the singulairty types that you get as the intersection is not enough. You need to remember their "mutual positions". (Alternatively, to remember the stratification of the space of smooth hypersurface-germs by the singularity types of the intersection.)

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