All Questions
Tagged with complex-geometry singularity-theory
56
questions
1
vote
0
answers
105
views
Enriques-Kodaira classification of minimal resolution of surface with quotient singularities
Let $X$ be a normal projective complex surface with at worst quotient singularities. Let $\bar{X}\to X$ be the minimal resolution. Further assume that $b_2(X)=1$ and $b_1(X)=b_3(X)=0$. Since quotient ...
- 189
1
vote
0
answers
50
views
Moduli space of curves away from singular subsets
Recently, I'm interested in the moduli spaces of curves in a (possibly noncompact) complex orbifold (resp. symplectic and almost complex) away from singularities.
More specifically, I'm interested in ...
- 275
7
votes
0
answers
354
views
Theories of manifolds w/ extra structure and singularities
Many different objects in mathematics can be described as manifolds with extra structure. Among the most famous examples of these are smooth manifolds, Riemannian manifolds, complex manifolds, and ...
- 141k
6
votes
0
answers
195
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Reference request: Automorphisms of $\mathbb C\{x,y\}$ which preserve the equation of the cusp, $x^3 - y^2$
In my research I encountered automorphisms of the ring of convergent power series
$$\varphi: \mathbb C\{x,y\} \to \mathbb C\{x,y\},$$
which preserve $f = x^3 - y^2$, i.e. $\varphi(f) = f$. I'm ...
- 1,141
0
votes
0
answers
118
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Is any singularity a subgerm of $(\mathbb{C}^n, 0)$?
I am studying singularity theory. I have often come across, in the literature, the sentence which says "let $(X,0) \subset (\mathbb{C}^n,0)$ be a singularity". Here a singularity is a ...
- 369
0
votes
0
answers
102
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Every locally free sheaf is Cohen-Macaulay on complex variety with at canonical singualities?
Suppose $X$ is a normal complex space with at most singularities, can we say any locally free sheaf on it is Cohen-Macauly?
Recall that a coherent sheaf $\mathcal{F}$ over $X$ is called (maximal) ...
- 295
5
votes
1
answer
368
views
Understand the proof that rational resolution is independent of the resolution
EDIT. Karl Schwede has given a different approach to solve this question, which is very clear. However, I still want to figure out how to deal with this problem along Ein's line, and waiting for the ...
- 295
2
votes
0
answers
132
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Jacobian ideal as primary idea;
Let $R = \mathbb{C}\{x_1, \dots,x_n\}$ be the ring of germs of analytic functions and let $f \in R$ be a homogeneus polynomial of degree $p$ such that $\sqrt{Jac(f)}=\langle x_1, \dots, x_{n-1}\rangle ...
1
vote
1
answer
144
views
Cohen-Macaulyness of Milnor algebra
Denote by $R = \mathbb{C}\{x_1, \dots, x_n\}$ the ring of germs of analytics maps at the origin in $n$ variables and let $f \in R$ such that $Sing(V(f))=V(x_1, \dots, x_{n-1})$ as sets. In addition, ...
8
votes
2
answers
844
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Gorenstein varieties: why the two definitions are equivalent?
There are two definitions of Gorenstein singularities
in the literature. Using Grothendieck's (or Serre's) duality, one
defines the "dualizing sheaf" an object $\hat K_M$ of derived category
...
- 9,157
12
votes
1
answer
615
views
Does a resolution of a rational singularity have rationally connected fibers?
A rational singularity is a singularity of a
complex variety $X$ such that for any
resolution $\pi:\; \tilde X\rightarrow X$ the
higher direct images $R^i\pi_*(O_{\tilde X})$
vanish for all $i>0$. ...
- 9,157
4
votes
0
answers
170
views
How to judge whether an orbifold is good
My own case comes from dynamic system on compact complex manifolds. To be precise, let $M$ be a compact complex 3-dimentional manifold, $W^c$ a holomorphic foliation of M with 1-dimentional uniformly ...
- 651
3
votes
0
answers
251
views
Monodromy group of the Milnor fiber of an ADE surface singularity
Let $X$ be a hypersurfaces in $\mathbb C^3$ defined by $f(x,y,z)=0$ with an ADE type singularity at $0$. Denote $\mu$ the Milnor number of the singularity.
On the one hand, we can fit $(X,0)$ into the ...
- 1,741
8
votes
1
answer
266
views
Is there a "minimal" Whitney stratification of a complex hypersurface?
Let $X\subset \mathbb C^n$ be a complex hypersurface (given by $F=0$ where $F$ is a polynomial). It is known then that $X$ admits a Whitney stratification. This is a decomposition of $X$ into smooth ...
- 14.1k
2
votes
0
answers
60
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Whitney stratification of a $\mathbb C^*$-invariant hypersuface in $\mathbb C^n$
Let $H\subset \mathbb C^n$ be an irreducible hypersurface invariant under a diagonal $\mathbb C^*$-action with positive weights ($H$ is given by a quasi-homogeneous polynomial). Consider the Whitney ...
- 14.1k