Skip to main content

Questions tagged [resolution-of-singularities]

Filter by
Sorted by
Tagged with
3 votes
1 answer
163 views

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 ...
Alex's user avatar
  • 365
3 votes
2 answers
241 views

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, ...
Rio's user avatar
  • 233
2 votes
0 answers
95 views

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 ...
David Hubbard's user avatar
2 votes
1 answer
180 views

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}\...
user41650's user avatar
  • 1,952
1 vote
0 answers
43 views

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 ...
user341068576's user avatar
2 votes
0 answers
79 views

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 ...
jg1896's user avatar
  • 2,763
1 vote
0 answers
115 views

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 $\...
singularity's user avatar
1 vote
0 answers
40 views

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 ...
Giovanny Snaider Barrera Ramos's user avatar
1 vote
0 answers
54 views

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}...
Igor Kotelnikov's user avatar
3 votes
0 answers
117 views

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 $k$ ...
Kostas Kartas's user avatar
2 votes
1 answer
303 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 ...
Shravan Patankar's user avatar
1 vote
0 answers
130 views

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
2 votes
1 answer
175 views

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
  • 3,301
2 votes
1 answer
326 views

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
  • 232
2 votes
1 answer
264 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 ...
H U's user avatar
  • 471
3 votes
0 answers
108 views

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
  • 1,361
3 votes
0 answers
128 views

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
3 votes
1 answer
276 views

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
1 vote
0 answers
126 views

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
  • 595
2 votes
1 answer
232 views

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
  • 21
0 votes
0 answers
69 views

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
  • 613
1 vote
0 answers
121 views

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
1 vote
0 answers
67 views

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
1 vote
1 answer
234 views

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
1 vote
1 answer
297 views

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
2 votes
0 answers
98 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 ...
Alexey Do's user avatar
  • 726
2 votes
0 answers
113 views

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
  • 7,646
1 vote
1 answer
182 views

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
  • 726
3 votes
1 answer
241 views

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
  • 1,627
3 votes
0 answers
118 views

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
  • 7,646
5 votes
0 answers
157 views

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
  • 1,823
2 votes
0 answers
169 views

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
  • 726
5 votes
1 answer
342 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
1 vote
0 answers
58 views

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
  • 263
3 votes
1 answer
198 views

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
1 vote
0 answers
144 views

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
3 votes
0 answers
228 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
332 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
  • 2,022
2 votes
1 answer
343 views

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
2 votes
1 answer
208 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
  • 141
2 votes
0 answers
206 views

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
  • 1,049
7 votes
1 answer
199 views

$2$-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
162 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
121 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
  • 3,362
8 votes
1 answer
260 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
  • 14k
1 vote
0 answers
229 views

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
  • 1,433
5 votes
1 answer
320 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
  • 9,421
4 votes
1 answer
414 views

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
  • 231
3 votes
0 answers
191 views

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
89 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

1
2 3 4 5