All Questions
Tagged with singularity-theory deformation-theory
25 questions
22
votes
5
answers
3k
views
Is a 'generic' variety nonsingular? Or singular?
I'd like to know whether there's some coherent meaning of 'generic' for which one can say that a 'generic' variety over an algebraically closed field $K$, say, is nonsingular or singular. We could ...
- 2,168
9
votes
0
answers
148
views
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
8
votes
1
answer
754
views
Why can you deform singularities in two dimensions but not in higher dimensions?
I've been trying to read this paper to understand deformations of surface quotient singularities. I'm particularly interested in when one can deform certain cyclic quotient singularities into other ...
- 3,290
7
votes
1
answer
770
views
Cohomology of tangent sheaf of a singular hypersurface
Let $X\subset\mathbb{P}^n$ be a hypersurface singular at finitely many points $p_i\in X$. We may assume that $X$ has ordinary singularities at the $p_i$'s.
Does there exists a formula, perhaps in ...
user91659
7
votes
0
answers
506
views
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
4
votes
2
answers
814
views
Are Du Val singularities smoothable?
I am interested in when a Du Val surface singularity is smoothable. By Du Val singularity, I mean (the germ of a) isolated double point surface singularity admitting a resolution by blowups of ...
- 205
4
votes
1
answer
433
views
Infinitesimal deformations of a singular projective surface
Let $X$ be a normal projective surface with just two singular points $x_1,x_2\in X$, where $X$ has rational quotient singularities.
Assume that both the singularities in $x_1$ and in $x_2$ admit a ...
user68440
3
votes
1
answer
327
views
Segre embedding and intersections by hyperplanes
Consider the Segre embedding $$ \mathbb{P}^2 \times \mathbb{P}^2 \to \mathbb{P}^8.$$ Denote by $V$ the image of the Segre embedding and by $B$ the locus of triples $(H_1, H_2, H_3)$ with $H_i \in H^0(\...
- 2,323
3
votes
1
answer
567
views
Deformations of quotient singularities
Let $Y$ be an affine scheme over a field of characteristic zero. Suppose we have a group $G$ acting on $Y$ and that the subset of $Y$ of points with non-trivial stabilizer is in codimension greater or ...
- 8,998
3
votes
1
answer
257
views
Bertini-type theorem for reducible schemes
Let $X \subset \mathbb{P}^n$ be a reducible, projective subscheme. Assume that $X$ is reduced (meaning that every local ring is reduced i.e., does not contain nilpotent element). Denote by $S_d$ the ...
- 2,032
3
votes
0
answers
215
views
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
1
answer
205
views
Deformation of isolated singularities and non zero divisors
Consider $f \in \mathbb{C}\{x_1,\dots,x_n\}$ such that $(V(f),0)$ has an isolated singularity.
Let $F \in \mathbb{C}\{x_1,\dots,x_n,t\}$ be a deformation of $f$ such that there exists some integer $m$...
2
votes
1
answer
89
views
Smoothings of isolated non-irreducible surface singularities
Let $(X,0)$ be a normal surface singularity. Suppose that it does not admit a smoothing.
Is it possible that there exists an isolated surface singularity $(Y,0)$ reduced near $0$ which is not ...
- 83
2
votes
1
answer
355
views
T^i functors are isomorphic for analytically isomorphic isolated singular points
I've been having trouble proving the following:
Let $B$ and $B'$ be local rings, essentially of finite type over $k$, having isolated singularities at the closed points. Suppose that they are ...
- 2,108
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.
...
- 761
2
votes
0
answers
263
views
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
1
answer
223
views
Does $\delta$-invariant give a sufficient condition for flatness for plane curve singularities?
Let $\pi:\mathcal{C}\to S$ be a morphism of schemes such that $\mathcal{C} \subset \mathbb{C}^2 \times S$ with the inclusion map commuting with the natural projection to $S$ and for all $s \in S$, $\...
- 2,126
1
vote
1
answer
178
views
Local discriminant variety
I'm looking for good (as simple as it is possible) reference for the local discriminant variety.
I need it in the following situation:
I have an unfolding $F: (\mathbb{K}^n \times \mathbb{K}^p, 0) \to ...
- 194
1
vote
1
answer
202
views
Obstruction map for local singularities via tangent (Andre-Quillen) cohomology
Let $R$ be a local singularity (for example $R=\mathbb{C}[[x_1, \ldots , x_n]]/I$) ring over $\mathbb{C}$. Let $\mathbb{L}_{R}$ be a cotangent complex of $R$, then one can define tangent (Andre-...
- 1,545
1
vote
0
answers
157
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
views
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
views
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