All Questions
Tagged with singularity-theory deformation-theory
11 questions with no upvoted or accepted answers
9
votes
0
answers
148
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Does every sequence of deformation of singularities eventually become equisingular?
Suppose we are over a field of characteristic zero and $f_i\colon X_i\to \mathrm{Spec}(R_i)$ $(i=1,2,\cdots)$ are flat families of singularities over DVRs. Assume that the generic fiber of $f_i$ is ...
- 131
7
votes
0
answers
506
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A general definition of an equisingular family of singular varieties?
This question is about the existence of a definition. I'm far from being an expert in the field in question I apologize in advance for any inaccuracies or stupid and wrong assumptions.
Let $X$ be a ...
- 7,799
6
votes
0
answers
388
views
Globalization of Brieskorn-Grothendieck resolution
Brieskorn (1970) showed that for semiuniversal deformation of rational double points surface singularities $X\to S$, there is a finite base change $S'\to S$, such that the new family $f:X\times_{S}S'\...
- 1,803
3
votes
0
answers
215
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cotangent complex for finite flat morphism and different ideal
Let a ring $A$, $F=A[X_{1},\dots X_{n}]$ and $B:= F/J$.
We suppose that we have a finite flat lci morphism $f:Spec(B)\rightarrow Spec(A)$.
To mesure the singularities of this map, Gabber-Ramero ...
- 3,472
2
votes
0
answers
91
views
Formal neighborhood of isolated singularity via DAG
I work over a field of characteristic $0$, denoted $k$. Let $f:\mathbf{A}^{d+1}\rightarrow\mathbf{A}^{1}$ have an isolated singularity at $0$, and let $\widehat{Z}$ denote the formal neighborhood of $...
- 121
2
votes
0
answers
136
views
quasi-ordinary singularities on a versal deformation?
Let $V$ be a variety over $\mathbb{C}$ and suppose $O$ is a singular point of $V$. Are there conditions on $(V,O)$ such that a versal deformation $W$ of $(V,O)$ has only quasi-ordinary singularities.
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- 761
2
votes
0
answers
263
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Do deformations of isolated hypersurface singularity naturally induce deformations of their divisors?
Let $0 \in V =(f=0) \subset \mathbb{C}^{n+1}$ be an affine variety with an isolated hypersurface singularity at the origin for $n \ge 3$.
Let $0 \in D=(x=f=0) \subset V$ be a divisor with only ...
- 909
1
vote
0
answers
156
views
Homogeneous deformation of isolated singularities
Let $f\in \mathbb{C}[x_1, \dots,x_n]$ be a homogeneous polynomial of degree $p$ and let $F \in \mathbb{C}[x_1, \dots,x_n,t]$ be a polynomial such that $F(x_1, \dots, x_n,0)=f$, for every $t_0$ we have ...
1
vote
0
answers
40
views
On Remmerts reduction
Let $(X,0)$ be a normal surface singularity. An let $\pi: \tilde{X} \to X$ be the minimal resolution. Now, we can apply a result of Oliveira (exploiting previous work by Laufer) and obtain a 1-...
- 1,409
1
vote
0
answers
250
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On regularity of flat families over a DVR
Let $k$ be an algebraically closed field of characteristic zero and $R$ a discrete valuation ring over $k$. Let $\pi:X \to \mathrm{Spec}(R)$ be a flat, projective morphism such that the generic fiber ...
- 1,981
1
vote
0
answers
154
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How do I check whether an orbifold admits deformations?
(Cross-post from math.stackexchange, where it has received no attention.)
Orbifolds $\mathbb{C}^2/\mathbb{Z}_n$, given by the action $(x, y) \mapsto (\zeta x, \zeta^{-1} y)$ with $\zeta$ a primitive $...
- 884