Questions tagged [resolution-of-singularities]

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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 ...
Shravan Patankar's user avatar
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Relative compactification without resolutions of singularities

Let $Y$ be a smooth proper variety over a field $k$, let $X$ be a smooth variety over $k$, $U\hookrightarrow X$ the complement of a strict normal crossing divisor and $\phi\colon U\to Y$ a map. By ...
user197402's user avatar
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Extending étale covers from the regular locus to a resolution of singularities

Let $X$ be a normal proper variety with rational singularities (or terminal if that is necessary) and $X_{\text{reg}} \to X$ the regular locus. Let $\pi : \tilde{X} \to X$ be a resolution of ...
Ben C's user avatar
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Lindelöf paper on meromorphic singularities

Does anyone know a digital link to the following paper, written by Ernst Lindelöf: "Mémoire sur certaines inégalités dans la théorie des fonctions monogènes, et sur quelques propriétés nouvelles ...
Felixson's user avatar
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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 ...
H U's user avatar
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Crepant resolution of quotient singularities

Let $G$ be a finite subgroup of $U(m)$ such that $G$ acts freely on $\mathbb C^m \setminus \{0\}$. If $\mathbb C^m/G$ has a crepant resolution, can we necessarily derive that $G \subset SU(m)$?
Adterram's user avatar
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Inverse image Weil divisor on a toric variety as a Cartier divisor

Let $X$ be a normal toric variety over an algebraically closed field and let $D$ be a torus invariant (prime) divisor. Assume $\pi\colon \tilde{X}\rightarrow X$ is a toric resolution of singularities ...
Boaz Moerman's user avatar
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Minimal resolution of singularities of surfaces

Let $X$ be a normal projective irreducible surface over an algebraically closed field $k$. Let $\pi\colon Y\to X$ be a birational morphism, such that $Y$ is a smooth projective surface, and assume ...
Jérémy Blanc's user avatar
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Question about the definition of variety in Kollár's book on resolution of singularities

In Kollár's book "Lectures on Resolution of Singularities" it is claimed in 3.8 page 125: "Our resolution is strong and functorial with respect to smooth morphisms" I would like to ...
Anette's user avatar
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Motives of resolutions of singularities

Suppose $X'$ is a resolution of singularities of a projective variety $X$ over a field $k$ of characteristic 0 that is functorial for smooth morphisms. How are the (mixed) motives of $X$ related to (...
Arna's user avatar
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How to determine the singlarity type (up to local analytic isomorphism) of a hypersurface surface singularity

Given a polynomial f(x,y,z), it defines a hypersurface in $\mathbb C^3$. I guess there is a classification of hypersurface singularity like Arnold normal form. I wonder given an explicite example of f,...
xin fu's user avatar
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Blow up of simply connected isolated singularity

Let $X \subset \mathbb{C}^n$ be a simply connected complex variety with a unique isolated singularity at $x\in X$. Let $\tilde{X}$ be the strict transform of $X$ under the blow-up $\mathrm{Bl}_x(\...
Serge the Toaster's user avatar
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Smooth affine variety as a symplectic resolutions

Given a smooth affine variety $X$ over $\mathbb C$ with an algebraic symplectic form $\omega$ and a finite group $G$ acting on $X$ by symplectomorphisms, then Is it true that $X$ is trivially a ...
Flavius Aetius's user avatar
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Whyt he pullback of the maximal ideal sheaf of the origin in $\mathbb{C}^2$ under blow-up of the origin is not torsion-free?

Maybe it is a silly question but i don't uderstand why the following statement is true: "Let X be a complex space and $\pi :Y \longrightarrow X$ be a proper modification of $X$. The pull back $\...
singularity's user avatar
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Beauville Exercise VII.7 (3)-A proof that $\kappa(X)\geq \kappa(Y)$ for $f\colon X\to Y$ surjective morphism of smooth projective varieties

Here $\kappa(X)$ denotes the Kodaira dimension of a smooth projective variety $X$. Question 1: I would like to solve Exercise VII.7 (3) from the Beauville book "Complex Algebraic Surfaces": ...
Federico Fallucca's user avatar
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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 ...
Alexey Do's user avatar
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An interpolation question for complex analytic hypervarieties

Suppose $D^{n} \subset \mathbb{C}^{n}$ is a small neighborhood of the origin, and $f, g: D^{n} \rightarrow \mathbb{C}$ are holomorphic functions. Moreover, $g$ is an arbitrarily small (i.e. $| f - g|_{...
nandi's user avatar
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Resolution of singularities of the resultant locus

We consider projective space of dimension $n$ as the parameter space of degree $n$ polynomials in one variable. Then, I am interested in resolving the singularities of the "resultant locus" $...
Asvin's user avatar
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Induced resolution of singularities

I am not a specialist in singularity theory but currently I have to touch resolution of singularities and I'd like to know whether I have understood Hironaka's theorem correctly. Let $k$ be a field of ...
Alexey Do's user avatar
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Resolution of conical singularities have even-only cohomology?

Considering a quotient singularity $\mathbb{C}^n/G,$ its crepant resolution $Y$ (i.e. having $c_1(Y)=0$) has rational cohomology supported in even degrees only. This holds for many other resolutions ...
Filip's user avatar
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Resolving the "wild" singularities of $\mathbb A^n/C_n$

Let the cyclic group on $n$ elements, $C_n$, act on $\mathbb A^n$ by permuting the co-ordinates (over a field $k$). If $n \neq 0 \in k$, we can resolve the singularities of $X = \mathbb A^n/C_n$ by ...
Asvin's user avatar
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Do quasi-excellent rings have a good constructive definition?

$\DeclareMathOperator\Sh{Sh}$Informally, a quasi-excellent ring is a Noetherian ring with a few technical extra regularity conditions that make it turns out to be the largest class of rings that allow ...
saolof's user avatar
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Trace formula for monodromy of Milnor fibrations

I am reading the paper A. Campo, Le nombre de Lefschetz d'une monodromie but I am stuck at several points, hope that someone can help me. Let $P:\mathbb{C}^{n+1} \longrightarrow \mathbb{C}$ be a germ ...
Alexey Do's user avatar
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1 answer
280 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 ...
Invariance's user avatar
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dimension of fibre of a generic point in an intersection of two sets

Let $M_m := (f_1, \cdots, f_m )$ be an algebraic map from $\mathbb{R}^n$ to $\mathbb{R}^m$ and $f_1^2,...,f_m^2$ are homogeneous polynomials of the same degree in $Q[x_1,...,x_n]$ . Similarly define $...
Pew's user avatar
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1 answer
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Searching for resolutions of generalized determinental varieties

I'm interested in studying a certain generalization of determinental varieties as defined here: https://en.wikipedia.org/wiki/Determinantal_variety To be more specific, I must first lay out a few ...
Kristaps John Balodis's user avatar
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Blow up singularities on curves

Let $p$ be a prime number and let $\bar{\mathbb F}_p$ be an algebraic closure of $\mathbb F_p$. Let $C$ be an irreducible singular projective curve over $\bar{\mathbb F}_p$. Let $P$ be a singularity ...
Yachen Liu's user avatar
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164 views

Birationally equivalent elliptic curves and singularities

I got the following cubic elliptic curve from some physical problem $$E_c(\mathbb{C}): w^2=4 z^3-zG_2-G_3,$$ where $G_2=3 \alpha ^2+\gamma$ and $G_3=\alpha ^3-\alpha \gamma -\beta ^2$ for known ...
DaveWasHere's user avatar
4 votes
1 answer
268 views

Do there exist linear relations between exceptional divisors

Let $X$ be an isolated, Gorenstein singularity of dimension at least $2$ and $\pi: \widetilde{X} \to X$ be a resolution of singularities. Let $E$ be the exceptional divisor and $E_1,...,E_r$ be the ...
Jana's user avatar
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1 vote
1 answer
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References on Namikawa-Weyl group

What are the most reasonable references on the definition of the Namikawa-Weyl groups and the first results about them? In particular, are there more recent (or more educational) texts than the ...
Vanya Karpov's user avatar
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1 answer
186 views

Preimage by birational maps

I am looking for an example (I guess that in complex projective space $\mathbb{P}^{n}$ is good) such that satisfy the following condition (in non trivial case, for this assume $X \neq \tilde{X}$): Let ...
Student85's user avatar
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plumbing description of resolution of ADE singularities

Let $G$ be a finite subgroup of $SU(2)$ and consider the quotient of the unit ball $B\subset \mathbb{C}^{2}$ by $G$. The result, denoted by $V$, has a boundary $S^{3}/G$ and has an ADE singularity at $...
user44651's user avatar
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7 votes
1 answer
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$2$-dimensional complete local normal domain with rational singularity that has exactly one exceptional curve

Let $(R, \mathfrak m)$ be a complete local normal domain of dimension $2$ with residue field $R/\mathfrak m$ algebraically closed and characteristic $0$. Assume Spec$(R)$ has rational singularity, let ...
Snake Eyes's user avatar
3 votes
0 answers
148 views

Does existance of crepant resolution of tangent space imply existance of crepant resolution globally in the algebraic setting?

Suppose $X$ is smooth proper algebraic $\mathbb C$-variety with algebraic action of a finite abelian group $G$. Suppose I know that $X/G$ (good geometric quotient) exists and it is normal Gorenstein ...
Mykola Pochekai's user avatar
1 vote
0 answers
101 views

Simultaneous Log resolutions for both varieties and divisors

Let $X$ be a normal variety and $D \subset X$ be a prime divisor which is also normal. It is well-known that we can take a resolution $f: W \to X$ of $X$ such that $$\DeclareMathOperator{\Supp}{\...
Li Yutong's user avatar
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8 votes
1 answer
243 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 ...
aglearner's user avatar
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1 vote
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Smooth normalization and blow-up of the exceptional locus

Let $n:\widetilde X\rightarrow X$ be the normalization of a complex (quasi-projective) variety $X$. Assume $\widetilde X$ is smooth, that $n$ is an isomorphism outside a smooth connected subvariety $Y\...
pi_1's user avatar
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5 votes
1 answer
277 views

Computing the invariants of ball quotient surfaces

The two-dimensional complex unit ball $B$ has group of biholomorphic automorphisms $PU(2,1)$. If $Γ$ is an arithmetic subgroup of $PU(2,1)$, the quotient $Γ\text{\\}B$ is an orbifold. Taking its ...
LeechLattice's user avatar
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Resolution of 3-fold quotient singularities

This is exercise 1.10 from Reid's Young person's guide to canonical singularites. Let $X=\mathbb{C}^3/ \mu_3$ where $\epsilon \in \mu_3$ acts by $$ (x,y,z) \to (\epsilon x, \epsilon y, \epsilon^2 z).$$...
WWK's user avatar
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When do crepant resolutions of quotients of Calabi-Yau varieties exist?

Suppose I have a Gorenstein variety $X$ over $\mathbb{C}$ with trivial canonical bundle, and the action of a finite group $G$ on $X$, which acts trivially on the canonical bundle. Question. When does ...
Josh Lam's user avatar
1 vote
0 answers
84 views

Partial crepant resolution in codimension 2

Let $\xi_5$ be a 5-root of the unity. We consider $\mathbb{C}^4/G$, where $G=\left\langle \sigma,\tau\right\rangle$, with $\sigma$ and $\tau$ the automorphisms given, respectively, by the following ...
Kovalevskaia's user avatar
5 votes
0 answers
169 views

Resolution graph of higher dimensional ADE singularities

I am looking for different configurations of the exceptional divisors arising from blowing up a higher dimensional ADE singularity (see p. 240 of this article of Bruns for a description of such ...
user43198's user avatar
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157 views

Normality of a blow up

Let $X$ be a quasi-projective normal $\mathbb C$-variety and $Y\subset X$ a smooth subvariety such that $X$ is normally flat along $Y$ (i.e. the normal cone of $X$ along $Y$ is flat over $Y$). Is the ...
pi_1's user avatar
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1 vote
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Blowing-up the exceptional locus of a (small) resolution

Let $f:\widetilde X\rightarrow Y$ be a proper morphism between smooth complex varieties which is birational unto its image $X=f(X)$. Assume the singular locus $W\subset X$ of $X$ is smooth and that $...
pi_1's user avatar
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0 votes
0 answers
167 views

Nice small resolution and normality of blow-up

Let $X$ be a complex variety whose singular locus is a smooth variety $Z$. Let $f:Y\rightarrow X$ be a small resolution of $X$ such that $f^{-1}(z)$ is smooth for any $z\in Z$ and $\dim(f^{-1}(z))$ is ...
pi_1's user avatar
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7 votes
2 answers
407 views

Symplectic resolutions amongst cotangent bundles

It is known that a generalized flag variety $X=T^*(G/P)$ is a (symplectic) resolution of singularities of its affinization $X^\text{aff}\mathrel{:=}\operatorname{Spec}(H^0(X,\mathcal{O}_X))$. In type ...
Filip's user avatar
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19 votes
1 answer
2k views

Perfectoid approach to resolution of singularities in char $p$

Since perfectoid techniques have built a bridge between char $0$ and char $p$ worlds, it is conceivable that they can be applied to resolution of singularities in char $p$ using their successful ...
Arna's user avatar
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0 answers
159 views

Factorization of birational maps in char $p$

So I was reading about the factorization result that any birational map between smooth varieties is composition of blow-ups and blow-downs with smooth centers. It is apparently true only in ...
user127776's user avatar
  • 5,607
1 vote
1 answer
114 views

Realizing a set as the image of a smooth map

Consider the following subset of $\mathbb{R}^2$: $S = \{ (x, y) \in \mathbb{R}^2 : |y| \leq |x|^{3/2} \}$ (See here for a plot on Wolfram Alpha.) The origin $(0, 0)$ is a kind of singular point of $S$....
Nicolas Boumal's user avatar
5 votes
2 answers
891 views

Cohomology of resolution of singularity

If $X,Y$ are smooth projective schemes, then if we have a surjection $f:X\to Y$, we have an injective map on étale cohomology, or more generally on any Weil cohomology (see https://mathoverflow.net/q/...
curious math guy's user avatar

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