# Tagged Questions

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### Relations between different but similar Milnor fibers

Let $f:\mathbb A^N\to\mathbb C$ be a holomorphic function and let $P$ be a critical point, i.e. $$P\in Z(\textrm df)\subset \mathbb A^N.$$ The Milnor fiber of $f$ at $P$ is the intersection of a ...
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### Can the Milnor number be used to resolve curve singularities?

Let $f(x,y)\in \mathbb{C}[x,y]$ define a curve $C$ which is singular at the origin. By successively blowing-up the origin, we can resolve the singularities of $C$. Of course to make sure this process ...
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### necessary and sufficient conditions for having an irreducible hyperplane section

Let $S \subset \mathbb{P}^3$ be a surface different to a cone. If $S$ is a smooth variety, then its generic hyperplane section is irreducible. I can imagine a similar result holds for a surface with ...
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### Relation of two definitions of 'Singular Point'

In my lecture we defined: For $(a,b)=P\in V(f)$, $P$ is a singular point if $\operatorname{mult}(f,P)>1$, where $\operatorname{mult}(f,P) = \max\{n\in\mathbb{N}\mid f\in\mathfrak{m}_P^{n}\}$, ...
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### Why is $P$ singular

This is from Shafarevich's 'Basic Algebraic Geometry 1': Let $P=(\alpha,\beta)\in X$, and suppose that the equation of $X$ is written in the form $f(x,y)=a(x-\alpha)+b(x-\beta)+g$, where $g$ is a ...
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### The multiplicity of $X$ at $x$ does not change when $X$ is cut by a generic hypersurface: what are those generic conditions?

Given an algebraic variety $X$ with a point $x \in X$, the multiplicity of $X$ at $x$ is defined as the multiplicity of the maximal ideal of $x$ in the local ring $\mathcal{O}_{X,x}$. In ...
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### Conditions for the number of degenerated fibers of a morphism to be finite.

I'm having trouble to find a theorem about this: Let's $F_z$ be a family of curves defined by $$F_z : F(x,y) + z = 0$$ where $F$ is an irreducible polynomial and $z \in \mathbb{C}$. My question is: ...
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### Do there exist double points on a surface in $\mathbb{P}_{\mathbb{C}}^3$ that are not rational?

The title explains it all. I'm familiar with the du val singularities on surfaces, also apparently known as rational double points (http://en.wikipedia.org/wiki/Du_Val_singularity). In ...
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### Singularities in (Elementary) Real Algebraic Geometry

I've taken an introductory course in algebraic geometry, and am currently studying Sumio Watanabe's book, "Algebraic Geometry and Statistical Learning Theory". In this book Watanabe gives a, possibly ...
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### Is it true that blowing up a quasi-affine variety at a nonsingular point never introduces new singularities?

If we let $M$ be a quasi-affine variety, is it true in general that the blowup of $M$ at a non-singular point $p$ does not introduce new singularities? I came across this statement in my reading, but ...
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### coefficients for getting a smooth equation

I'm studying the equation $$x_0q_0 + x_1q_1 + x_2q_2 = 0$$ where $q_i$ is a homogeneous polynomial of degree two in the variables $x_0,\ldots,x_5$. I would like to have some simple choices for the ...
Suppose that $Y$ is a normal variety such that its canonical class $K_Y$ is $\mathbb{Q}$-Cartier. , and let $f:X \to Y$ be a resolution of the singularities of $Y$. Then $$K_X= f^*(K_Y)+\sum_i ... 0answers 66 views ### Topology of singularities In the theory of surface singularities, it is well known that the topology of singularities is determined by its resolution dual graph. What`s the meaning of the topology of singularities? Here the ... 1answer 93 views ### How many types of surface singularities multiplicity two exist? All varieties are over \mathbb{C}. Let S be a reduced algebraic surface in \mathbb{P}^3 with a singular point p of multiplicity two. The question is local so we reduce to S \subset ... 0answers 75 views ### Puiseux series and Resolution of Singularities I have a very basic knowledge of algebraic geometry(no schemes!), and am trying to study the resolution of singularities. So the Newton's method gives us a Puiseux series parametrizing the branches of ... 0answers 43 views ### Algorithm for checking that a singularity is an ordinary double point Let a complete intersection d-fold singularity be cut out by k polynomials \{f_1,\ldots,f_k\} in \mathbb{C}^{k+d}. Is there a relatively simple algorithm to check whether this is analytically ... 0answers 48 views ### How do I check whether an orbifold admits deformations? Orbifolds \mathbb{C}^2/\mathbb{Z}_n, given by the action (x, y) \mapsto (\zeta x, \zeta^{-1} y) with \zeta a primitive n^\text{th} root of unity, admit smooth deformations. This is because ... 1answer 72 views ### Local rings and classifying singularities My query is a little vague, but I'll try to be as concrete as possible. Is there some sense in which the local ring of an algebraic variety (or more general complex space) at a point depends only on ... 0answers 63 views ### Question on definition of 3-fold cDV singularities A 3-fold compound Du Val singularity of type A_{k} \ (D_{k}, E_{k}) is defined as a singular point on a 3-fold which is locally given by$$ F(x,y,z,w)=f(x,y,z)+wg(x,y,z,w)=0,  where $g(x,y,z,w)$ ...
I quote from my lecture: Let $X$ be a topological space (think of $X=\mathbb{C}^n$ with the classical topology), $p\in X$, $A,B\subseteq X$. Then $A\sim B$ if there exists an open subset ...