# Questions tagged [resolution-of-singularities]

The resolution-of-singularities tag has no usage guidance.

232
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### Image, upto direct summands, of derived push-forward of resolution of singularities

Let $\mathcal C$ be a full subcategory (closed under isomorphism also) of an additive category $\mathcal A$. Then, $\text{add}(\mathcal C)$ is the full subcategory of $\mathcal A$ consisting of all ...

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### Question about surface singularities

Throughout, $X$ will be a projective surface. I am looking for examples of the following surface singularities,
I) A rational singularity that is not quotient. Obviously, it has to be non-Gorenstein, ...

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### Singularities of curves over DVRs with non-reduced special fibre

Let $R$ be a complete DVR of mixed characteristic with fraction field $K$ of characteristic $0$ and residue field $k$ of characteristic $p>0$. Suppose that $\mathcal{X}$ is a normal $R$-curve such ...

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### Perfect complexes of plane nodal cubic curve

Let $C\subset\mathbb{P}^2$ be a plane nodal cubic curve with a unique singular point $O$ at the origin. Then I consider its normalization, denoted by $\widetilde{C}$ and let $\pi:\widetilde{C}\...

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### Do local and global symplectic resolutions have same monodromy?

Yoshinori Namikawa associates a Weyl group $ W $ to any symplectic affine complex variety $ X $ with good $ \mathbb{C}^* $-action. He provides a semi-explicit description of $ W $, which requires ...

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### Can Coulomb branches have symplectic resolutions?

My question is about Coulomb branches of a $3D$ $\mathcal{N}=4$ supersymmetric gauge theory, in the sense of Bravermann, Finkelberg and Nakajima Towards a mathematical definition of Coulomb
branches ...

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115
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### Log resolution of ideal and associated dual graph

Let $(X,0)$ be a complex surface germ with an isolated singularity and $I$ be an $\mathfrak{m}\text{-primary}$ ideal (contains a power of the maximal ideal $\mathfrak{m}$) of the local ring $\...

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### Verdier (w) condition implies the $w_f$ condition when the restriction of $f$ in each stratum is a submersion?

Let $X\subset\mathbb{R}^n$ be and let $\Theta=(X_\beta)_{\beta\in I}$ a Verdier stratification for X. Let $f:X\rightarrow\mathbb{R}$ be a polynomial function, such that $f_{|_{X_\beta}}$ is submersion ...

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### How can I calculate the derivative of an integral with respect to a parameter if Leibniz's formula gives a divergent integral?

We are working on the problem related to a magnetic field in an axially symmetric magnetic plasma trap. Let's express the vector potential through the magnetic flux function
\begin{gather}
\label{1:01}...

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### Semi-stable model over a totally ramified extension

Notation: Let $R$ be a DVR, $K=\text{Frac}(R)$ and $k=R/\mathfrak{m}$. Given an $R$-scheme $X$, write $X_K=X\times_{R} K$ for the generic fiber and $X_k=X\times_R k$ for the special fiber.
Suppose $k$ ...

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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 ...

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130
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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 ...

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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 ...

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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 ...

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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 ...

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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 $G \subset SU(m)$?

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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 ...

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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 ...

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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 ...

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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 (...

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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,...

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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(\...

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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 ...

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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 $\...

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297
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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":
...

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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 ...

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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" $...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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### 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 ...

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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 $...

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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 ...

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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 ...

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### 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 ...

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### 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 ...

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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 ...

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### 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 ...

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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 $...

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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 ...

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### 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 ...

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### 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}{\...

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### 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 ...

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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\...

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### 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 ...

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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
$$ (x,y,z) \to (\epsilon x, \epsilon y, \epsilon^2 z).$$...

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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 ...

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### 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 ...