# Tagged Questions

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### de jong's alteration theorem for families

What is the current status of de Jong's smooth alteration theorem for a family of schemes? His 1997 paper shows that given any family of curves $X/S$ with $S$ of finite type (and, say, local) over a ...
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### Resolution of “nice” and zero-dimensional singularities on a surface

Assume I have a singular algebraic surface $X$ over an algebraically closed field (characteristic zero if you want) which is singular in a finite set of points. I am looking for a condition as to the ...
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### The exceptional locus of a minimal resolution of singularities

Let X be a surface. (A surface is an excellent integral normal separated 2-dimensional scheme.) Let $\psi:Y\longrightarrow X$ be a minimal resolution of singularities and let $E$ be an irreducible ...
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### Stable singularities of smooth map $\mathbb R^3\to \mathbb R^4$

Does anybody know any classification of stable singularities of smooth map $f:\mathbb R^3\to \mathbb R^4$? It is clear that there are singularities which look like intersection of 2 (or 3 or 4) ...
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### Du Val singularity of type G=A,D,E and “small” representations of G

We all know that a simple singularity $W_G(x_1,x_2,x_3)=0$ of type G=A,D,E has the following nice deformation involving the Cartan subalgebra $\mathfrak{h}$ of the Lie algebra $\mathfrak{g}$ of $G$. ...
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### Tangent cones to Severi strata

Let $\mathbb{C}[[x,y]]/f(x,y)$ be a reduced plane curve singularity. The base of a versal family can be taken to be (an open subset in) $\Lambda = \mathbb{C}[x,y]/(f,\partial_x f, \partial_y f)$; the ...
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### Brieskorn's proof of a theorem by Milnor about the Milnor number

I am looking for a reference or short explanation of a proof by E. Brieskorn. In his famous work "Singularities of complex hypersurfaces" Milnor proves that the (nowadays called) Milnor Number (in ...
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### Transversals to singular subvarieties

Say $\mathbb{C}^d \subset Y^{N-k} \subset \mathbb{C}^N$ are closed imbeddings of complex analytic subvarieties of the indicated dimension, $Y$ is not smooth. At a point $y \in Y$, a generic, ...
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### Can one obtain surfaces with interesting invariants as resolutions of singular surfaces?

(Perhaps a not very well defined question) Let $(S_t)_t$ be a (flat) family of compact complex surfaces. Assume the generic member is smooth while $S_0$ has isolated singularities. As the simplest ...
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### An elementary question in singularities

The following problem came up in something I am working on. It has a really elementary statement but I couldn't crack it in a couple of hours of thinking about it. It isn't clear to me if I am being ...
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### Singularity structure of integrals of rational functions

Suppose I have a convergent integral of the form $\int_0^1dx_1\dots\int_0^1 dx_n \frac{P(x_i)}{Q(x_i)}$, where $P$ and $Q$ are polynomial functions of $n$ nonnegative real variables $x_i$. Let the ...
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### on the relative conductor of curve singularity and quotient of ideals

Let $R$ be the local ring of a complex curve singularity. (Can assume the singularity planar, the ring locally analytic or formal.) Let $\bar{R}$ be the normalization, let $R\subset R'\subset \bar{R}$ ...
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### Higher dimensional nodes

A node on a curve is a singular point that locally looks like the intersection of two lines. I think the precise way to say this is that $p \in X$ is a (closed?) point on a scheme $X$ (of finite type ...
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### complex singularity exponent, lct

Hi everybody, I have a question about log canonical thresholds / complex singularity exponents. If I understood well, this invariant sees more things than the multiplicity, for example, the cusp in ...
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### Is resolution of singularities effective?

Suppose I have a singular projective variety defined by some homogeneous equations in complex projective space. Is the resolution of singularities effective? That is, do I actually know which smooth ...
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### Are orbifold singularities canonical?

This is a direct consequence of my previous question: Extending group actions on varieties In his answer, inkspot said that group actions can be extended if the variety has ample canonical class and ...
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### Does any one understand the details of M Kazarian's work in enumerative geometry of $\mathbb{C}\mathbb{P}^2$ ?

I wanted to know if anyone understood the details of the paper "Multisingularities, cobordisms, and enumerative geometry" available at the site http://www.mi.ras.ru/~kazarian/. In particular does ...
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### Which presentations of (non)planar algebras give rise to knots?

Reidermeister's theorem states that the set of knots, modulo ambient isotopy, is isomorphic to the planar algebra generated by crossings, modulo Reidemeister moves. This planar algebra presentation is ...
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### How to enumerate curves with a singularity once you know the corresponding Thom Polynomial?

Does any one know how to go from "Thom polynomials" to "Enumerating curves". I believe there is a relation between the two, but I don't know how to go about it. Let me make the question more concrete....
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The questions first: What is the proof of resolution of singularities that you know? Why am I asking?: There are a number of proofs of resolution of singularities of varieties over a field of ...
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### A necessary and sufficient condition for a curve to have an $A_k$ singularity.

Hi Does any one know of a necessary and sufficient condition for a curve to have a singularity of type A_k. More precisely, a curve f=0 has a singularity of type A_k at a point, if there exist local ...
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### Small neighborhoods of singularities on varieties

In Singular points of complex hypersurfaces, John Milnor proves the following theorem: Let $x \in V$ be a point on a variety $V$ in $\mathbb{R}^n$ or $\mathbb{C}^n$. Assume $x$ is either a smooth ...
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### How/where are semi-log resolutions used?

In the paper, by János Kollár there is problem 19 (page 8). It is one more strict resolution. A resolution that leaves untouched the semi-simple-normal-crossings singularities of pairs. My question ...
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### What strict resolutions of singularities are needed?

Suppose we have a collection, $S$, of singularities types and consider a resolution of singularities (this is: a proper birrational morphism $Y\rightarrow X$ such that Y only contains singularities of ...
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### Log resolutions of linear series

Let $X$ be a complex normal projective variety, let $|L|$ be a non empty linear series on $X$ and let $b(|L|)$ be its base ideal. Suppose $f:X'\rightarrow X$ is a log resolution of the ideal $b(|L|)$. ...
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### Is the desingularization of a normal variety with only quotient singularities projective

The base field will be the field of complex numbers. I have a slightly technical problem concerning the resolution of singularities of a certain variety. Basically, I want to to know if it is ...
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### The link of a singular quintic hypersurface in CP^4

Given a family of quintic hypersurfaces in $\mathbb{CP}^4$ by $x_1^5+x_2^5+x_3^5+x_4^5+x_5^5+(5+\epsilon)x_1x_2x_3x_4x_5$ we get a singular variety for $\epsilon=0$ with 125 singular points. I know ...
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### Is there an obvious way for showing singularities are quotient?

I'm stuck on a technicality concerning singularities. Basically, I have to show that the singularities of a $\mathbf{certain}$ normal projective variety over $\mathbf{C}$ are rational. (I won't ...
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### When singular points of a reduced scheme are not dense in it?

A stupid AG question: could singular (Zarisky) points be dense in a reduced (Noetherian) scheme $S$? If yes, which 'standard' restrictions on $S$ could ensure that this does not happen? For example, ...
Let $Y$ be a normal projective surface, let $X$ be a smooth projective surface and let $\pi:Y\longrightarrow X$ be a finite morphism. Why are all singularities of $Y$ cyclic quotient singularities? ...