All Questions
Tagged with singularity-theory ac.commutative-algebra
49 questions
27
votes
2
answers
1k
views
Limit of a series of singularities
The $A_\infty$ and $D_\infty$ plane curve singularities have defining equations $x^2=0$ and $x^2y=0$. These equations are "clearly" natural limiting cases of the equations for $A_n$ singularities $x^...
- 5,846
15
votes
4
answers
1k
views
What formal properties should resolution of singularities have?
If I were going to propose a new construction as a "replacement for resolution of singularities", what properties would my replacement have to have? [I am going to do no such thing -- this is purely ...
- 5,846
13
votes
1
answer
3k
views
Factoriality: local or global?
Let $X$ be an algebraic variety. I have read the following definitions:
$X$ is factorial if every Weil divisor on $X$ is Cartier.
$X$ is locally factorial if all its local rings are unique ...
- 884
11
votes
1
answer
865
views
Valuation of an ideal in a two-dimensional regular local ring
Let $f,g$ be two coprime elements in the ring $K[[x,y]]$, with $K$ a field.
What is the smallest integer $n$ such that the inclusion of ideals $$(x^n)\subset (f,g)$$ holds in $K[[x,y]]$? Can we ...
- 111
10
votes
1
answer
981
views
Factoriality vs $\mathbf{Q}$-factoriality for threefolds hypersurfaces with isolated singularities
Let $X \subset \mathbf{P}^4$ be a complex threefold hypersurface with isolated singularities.
We denote as usual by $\textrm{Cl}(X)$ the group of Weil divisors modulo linear equivalence and by $\...
- 66.3k
10
votes
0
answers
573
views
Singularities arising from the Minimal Model Program (an algebraic point of view)
I will start the story by the end:
Is there some characterization of (some of) the singularities arising from the Minimal Model Program (canonical, terminal, log-...) in terms of commutative algebra ?...
- 987
9
votes
0
answers
2k
views
Jacobian ideals reference
Suppose that $f : X \to V$ is a flat equidimensional (of dimension $h$) morphism of schemes of finite type and $V$ is excellent (or a variety) For this one can formulate something called the Jacobian ...
- 20.5k
8
votes
1
answer
493
views
General conditions for normality of blow-up
Let $X$ be an integral, affine, normal complex surface. I am looking for conditions on zero-dimensional closed subschemes $Z$ in $X$ such that the reduced scheme associated to the blow-up of $X$ along ...
- 2,032
8
votes
0
answers
398
views
A criterion for rational singularities in mixed characteristic
Let $R$ be a mixed characteristic discrete valuation ring with perfect residue field and $f:X \to \mathrm{Spec}(R)$ a flat proper morphism.
If the generic fibre of $f$ is smooth and the special fibre ...
- 10.5k
7
votes
1
answer
553
views
Relationship between Hilbert-Samuel multiplicity and polar multiplicity
Let $f \in \mathbb{C}[[x,y]]$ be the germ of an isolated plane curve singularity. Then the Hilbert-Samuel multiplicity $e_f$ of $f$ is given as follows:
$$e_f = \lim_{s \to \infty}\frac{1}{s} \cdot \...
7
votes
0
answers
518
views
An elementary question in singularities
The following problem came up in something I am working on. It has a really elementary statement but I couldn't crack it in a couple of hours of thinking about it. It isn't clear to me if I am being ...
- 3,758
6
votes
1
answer
175
views
Is there an algorithm to find out the number of small solutions to a polynomial equation, when we vary all the coefficients?
Let $\Phi (z,t)$ be a polynomial given by
$$ \Phi(z,t) := z^n + A_{n-1}(t) z^{n-1} + \ldots + A_1(t) z + A_0(t).$$
Assume that $\Phi(0,0) =0$. It is a fact that a solution $z(t)$ of the equation
$$ \...
- 3,245
5
votes
1
answer
359
views
Computations of divisor class monoids
Let me first recall some definitions from the very first pages of Bourbaki, Commutative Algebra, Chapter 7, "Divisors".
Let $A$ be a (commutative) domain, $K$ its field of fractions. A ...
- 26.1k
5
votes
1
answer
546
views
For what varieties do we have results on the category of singularities?
Let $X$ be a singular variety. Define the (triangulated) category of singularities (as in Orlov's paper)
as the Verdier quotient of the derived category of coherent sheaves on $X$ modulo the full ...
- 1,423
5
votes
0
answers
494
views
ramification locus for finite morphism and Abhyankar's Lemma
I want to ask given a finite morphism between projective varieties $f:X\rightarrow Y$.
What is exactly the ramification locus $\Delta(X/Y)$. If $X$, $Y$, $f$ are smooth, then I can more or less ...
- 623
4
votes
1
answer
282
views
Is pseudo-rationality preserved by etale morphisms?
Let $f: Y \to X$ be an etale morphism of schemes.
If $X$ has pseudo-rational singularities then does $Y$ also have pseudo-rational singularities?
For the definition of pseudo-rational see, for ...
- 10.5k
4
votes
1
answer
165
views
The volume around a singular isolated root when equalities are loosened
Suppose I have a system of polynomial equations in $n$ real variables $f_i(x_1,\ldots,x_n)=0$, $i=1,\ldots,m$, such that $0$ is an isolated solution. Now I replace each of the equations with a double-...
- 5,971
4
votes
0
answers
79
views
Possible number of zeros of a stable perturbation of a germ $(\mathbb{R}^n, 0) \to (\mathbb{R}^n, 0)$
Let $f:(\mathbb{R}^n, 0) \to (\mathbb{R}^n, 0)$ be an analytic germ. Assume that it has isolated zero at 0, that is, $f^{-1}(0)=\{0\}$, what is more, assume that the dimension of the local algebra* $Q(...
- 73
4
votes
0
answers
112
views
On the milnor number of analytic germ map
If $f:(\mathbb{C}^n,0) \to (\mathbb{C},0)$ is an analytic germ with an isolated singularity, then the Milnor number of $f$, denoted by $\mu(f)$ can be defined as $\dim_{\mathbb{C}} \mathcal{O}_n/\text{...
- 558
4
votes
0
answers
168
views
Can nonflat deformations of singularities always produce Cohen-Macaulay rings?
To make the question in the title precise, let me phrase it like this. Consider a complete local ring
$$ A := \mathbb{C}[[x_1, \dotsc, x_n]]/(f_1, \dotsc, f_m) $$
and, for definiteness, assume that $...
- 2,663
4
votes
0
answers
897
views
A strong form of implicit function theorem (what happens when the derivative is degenerate?)
(this can be considered as some ad)
Consider the system of equations $F(x,y)=0$. (Here $x$, $y$ are multi-variables. The equations are over a local ring. e.g. polynomial/analytic/formal/$C^\infty$ ...
- 2,232
3
votes
3
answers
681
views
on the relative conductor of curve singularity and quotient of ideals
Let $R$ be the local ring of a complex curve singularity. (Can assume the singularity planar, the ring locally analytic or formal.) Let $\bar{R}$ be the normalization, let $R\subset R'\subset \bar{R}$ ...
- 2,232
3
votes
2
answers
396
views
Cohen-Macaulay Representations
I came across the book "Cohen-Macaulay Representations" by Graham J. Leuschke and Roger Wiegand, and now I'm wondering if this is an active area of research.
If yes, then
what are some of ...
- 839
3
votes
1
answer
267
views
Singularities of contractions of extremal faces
Let $(X, \Delta)$ be a (projective) klt pair (say over $\mathbb{C}$, but I am also interested in fields of positive characteristic) and $f: X \to Z$ the contraction associated to a $(K_X + \Delta)$-...
- 10.5k
3
votes
4
answers
1k
views
Matrix factorization categories for ADE singularities
What is known about the matrix factorization categories of singularities of type ADE? Any references on this would be greatly appreciated.
Background: For ADE singularities, see for example this. For ...
- 21k
3
votes
0
answers
92
views
Solutions of the vector field $D=A\frac{\partial}{\partial X}+B\frac{\partial}{\partial Y}$ with $A,B\in k[[X,Y]]$
I am trying to understand the article Reduction of Singularities of the Differential Equation $Ady=Bdx$ by Arno van den Essen.
Let $A,B\in k[[X,Y]]$ be formal power series and $D$ the vector field $A\...
- 31
3
votes
0
answers
992
views
Definition of Q gorenstein variety
I have a question about the definition of Q-Gorenstein variety.
I saw a definition of Q-Gorenstein variety：for a normal variety $X$, it's Q-Gorenstein if the canonical divisor is Q Cartier. I wonder ...
- 623
3
votes
0
answers
131
views
Classification of faithfully flat morphisms between formal power series
Let $\mathbb{C}[[z_1,\dots,z_n]]$ denote the algebra of formal power series.
I am interested in faithfully flat morphisms
$$Spec(\mathbb{C}[[z_1,\dots,z_m]])\to Spec(\mathbb{C}[[z_1,\dots,z_n]]),\, m\...
- 21.8k
3
votes
0
answers
342
views
Hypersurfaces with Gorenstein singular loci
Recall that a hypersurface $D$ in a complex manifold $X$ is called a free divisor if the Lie algebroid $\mathcal{T}_X(-\log D)$ of vector fields tangent to $D$ is locally free. This condition is ...
- 126
2
votes
1
answer
138
views
Solvability of derivation Lie algebras of local finite-dimensional commutative algebras
Let $A$ be a finite-dimensional local commutative algebra (with one) over a characteristic zero field $k$. Is it true that the Lie algebra $\operatorname{Der}_k(A)$ of $k$-derivations of $A$ is ...
- 18.4k
2
votes
1
answer
304
views
Normal forms of ADE singularities
Given a surface $X:f(x,y,z)=0\subset \mathbb{A}^{3}_{\mathbb{C}}$ with only ADE singularities, how does one determine the correct singularity type of $X$ by computing the normal forms?
Does a similar ...
- 481
2
votes
1
answer
327
views
Krull dimension of the smooth locus
Let $R$ be a normal complete local domain of dimension $n \geq 4$. Does there exist a prime ideal $\mathfrak{p}$ of height $\dim(R) - 1$ such that $R_{\mathfrak{p}}$ is a regular local ring? In ...
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
259
views
Scheme-theoretic image and delta-invariants
Let $(X,o)$ be an affine, isolated, normal, Gorenstein singularity. Let $f$ and $g$ be two morphisms from $\mbox{Spec}(\mathbb{C}[[t]])$ to $X$ (also known as formal arcs) such that the closed point ...
- 1,593
2
votes
0
answers
108
views
Deformation to normal cone of the exception divisor of a log-resolution
I am reading the paper Iterated vanishing cycles, convolution, and a motivic analogue of a conjecture of Steenbrink due to G. Guibert, F. Loeser, and M. Merle. The main tool, like a lot of papers in ...
- 893
2
votes
0
answers
134
views
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 ...
2
votes
0
answers
288
views
Torsion freeness of direct image of structure sheaf?
I have the following question.
Let $f:X\rightarrow Y$ be a surjective projective morphism between smooth projective varieties.
I learned that if $\dim Y=1$, then $R^if_*\mathcal O_X$ is torsion free $\...
- 623
2
votes
0
answers
300
views
How to make a product of polynomials irreducible?
Let $p(x,y),q(x,y)\in \mathbb{Q}[x,y]$. Assume that $p(0,0)=q(0,0)$. Is it generally possible to find a polynomial $r(x,y)\in\mathbb{Q}[x,y]$ irreducible in $\mathbb{Q}[x,y]$ such that $\mathbb{R}[x,y]...
- 29
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
1
answer
505
views
Family of curve singularities whose generic members are smooth
Let $f: (X,x)\rightarrow (\mathbb C,0)$ be a deformation of a curve singularity $(X_0,x)$, and let $f: X \rightarrow T$ be a sufficiently small representative. Assume that $(X,x)$ is reduced and pure ...
- 21
2
votes
0
answers
102
views
semicontinuity of the conductor defined by Temkin
We say a principal pair $(X,\mathcal{I})$ where $X=Spec(A)$ is affine scheme and $\mathcal{I}=\tilde{I}$ where $I\subset A$ is a principal ideal generated by $\pi$ wich is a non zero divisor.
For a ...
- 3,472
1
vote
1
answer
408
views
Jacobian ideal regular sequence
Let $f(x,y) \in \mathbb C\{x,y\}$ be a holomorphic function-germ at zero. Let $f_x, f_y$ denote its partial derivatives. What is the proof of the following statement?
If the $\mathbb C$-algebra $\...
- 81
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
1
answer
149
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, ...
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
581
views
generalization Abhyankar's lemma
This question is related to a question I already asked on MO (smooth quotient out of a singular variety?), but I realized later that the hypotheses where not precise enough in my former question.
Let ...
- 7,320
0
votes
2
answers
296
views
Hochster-Roberts Theorem reciprocal
Given a Cohen-Macaulay ring $R$ over a field of characteristic zero and $G$
a reductive algebraic group acting on $R$, then the ring of ivanriants $R^G$
is also Cohen-Macaulay. This is known as ...
- 1,595
0
votes
1
answer
286
views
A condition on isolated singularity
Suppose $F: {\mathbb C}^N \to {\mathbb C}$ defines a singularity at the origin (for simplicity one can assume that $F$ is a quasi-homogeneous polynomial). Suppose it is nondegenerate, i.e., $dF(z) = 0$...
- 1,207
0
votes
0
answers
135
views
On resolution of singularities over an Artin ring
For a locally noetherian scheme $X$, Grothendieck conjectured that if $X$ is quasi-excellent then there is a proper birational map $Y \to X$ s.t. $Y$ is regular.
We now fix an Artin ring $R$ whose ...
- 519