3
votes
0answers
32 views

Why is this Milnor fiber homeomorphic to a cylinder?

Let $f:(\mathbb C^2,0)\to (\mathbb C,0)$ be a holomorphic function with a critical point at the origin. Let us denote by $X_0$ the fiber $f^{-1}(0)$, in which, as we said, the point $0\in X_0$ is a ...
0
votes
0answers
19 views

Branch locus of a projection of algebraic set

Let $X$ a algebraic cone in $\mathbb{C}^n$ with $\dim_0 X=p$. def.: if $f:A\to B$ is smooth map between smooth manifolds, then $br(f)$ is the points $x$ that $df_x$ is not surjective. def.: if $\pi: ...
3
votes
0answers
26 views

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 ...
2
votes
0answers
42 views

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 ...
2
votes
0answers
49 views

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 $\operatorname{mult}(f,P) = \max\{n\in\mathbb{N}\mid f\in\mathfrak{m}_P^{n}\}$, ...
3
votes
1answer
40 views

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 ...
2
votes
0answers
54 views

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 ...
2
votes
1answer
33 views

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: ...
1
vote
0answers
49 views

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 ...
2
votes
1answer
67 views

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 ...
4
votes
0answers
84 views

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 ...
1
vote
1answer
49 views

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 ...
3
votes
1answer
56 views

Discrepancy non log canonical singularities

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 ...
1
vote
0answers
61 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 ...
4
votes
1answer
75 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 ...
2
votes
0answers
71 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 ...
2
votes
0answers
39 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 ...
2
votes
0answers
44 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 ...
2
votes
1answer
69 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 ...
1
vote
0answers
54 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 $g(x,y,z,w)$ ...
2
votes
2answers
168 views

The notion of a germ in singularity theory

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 ...