Questions tagged [resolution-of-singularities]
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245 questions
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Extend algebraic morphism to a compactification with normal crossing boundary
Suppose $X$ and $Y$ are smooth algebraic variety over a char $0$ field $k$, and $f:X\to Y$ a morphism. I want to ask whether there exists compactifications $\bar X$ and $\bar Y$ such that $\bar X\...
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Compactification of smooth varieties with normal crossing boundary
I see in this paper, page 46, the second sentence of 4.1, that every smooth variety over a characteristic $0$ field can be embedded into a proper smooth variety with normal crossing boundary, and the ...
- 785
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57
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Discrepancy of general element of linear system
Let $X$ be a normal scheme and $|D|$ a linear system on $X$.
In "Singularity of Minimal Model Program" by Janos kollar p249, it says,
If $X$ is a variety over $\mathbb{C}$, and $E_j$ ...
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151
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Compactification of the Jacobian of singular curves via parabolic modules
I would like to better understand a certain compactification of the Jacobian variety of a singular algebraic plane curve as described in Cook's Ph.D. 1993 thesis Local and Global Aspects
of the Module ...
- 1
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88
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Equivariant resolution of singularity making a pullback of a line bundle admit a root
I am considering the following situation. Let $X$ be an irreducible normal projective variety with an action of a linear algebraic group $H$, and we have a $H$-equivariant line bundle $L$ over $X$. We ...
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Blow up of terminal singularity and canonical singularity
A normal singularity $(X,x)$ over a field $k$ is terminal (resp. canonical) if
$(i)$
it it is $\mathbb{Q}$-Gorenstein. and
$(ii)$For any resolution of singularity $F:Y\rightarrow X$,
$K_Y-f^*K_X>...
- 328
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487
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Variety without a compactification whose complement is smooth
Let $X$ be a smooth, separated complex algebraic variety. By Hironaka, there exists a compactification $j : X \to \bar{X}$ of $X$ so that $\bar{X} \setminus X$ is a simple normal crossings divisor.
Is ...
- 813
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Explicit expression of simultaneous resolution of semi-universal deformation of ADE singularity
Denote the semi-universal deformation of ADE singularity by $\mathcal{Y}\to\mathfrak{h}^{\mathbb{C}}/W$, where $\mathfrak{h}^{\mathbb{C}}$ is the complex Cartan algebra of root system of type ADE and $...
- 93
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About the definition of cDV singularity
M. Reid defines cDV singularity as follow in his paper "CANONICAL 3-FOLDS"
A point $p\in X$ of a 3-fold is called a compound Du Val point if for some section H throgh $P$, $P\in H$ is a Du ...
- 328
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Finiteness of rational double point
Let $(R,\mathfrak{m
})$ be a three dimensional complete local ring over a field $k$ of arbitrary characteristic and let $f\in R$ and $R/f$ is a rational double point.
My question is as follows.
Are ...
- 328
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196
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Terminal singularities of fibers vs total space
Suppose $f\colon X \to Y$ is a flat map of complex varieties (or more generally DM stacks?). Suppose every fiber has at most terminal singularities and that $Y$ is smooth. Under what conditions is it ...
- 43
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151
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Examples of small resolutions in dimension 4 and higher
I have seen numerous examples of 3-folds admitting small resolutions. Are there similar examples in higher dimension? In particular, I am looking for examples of singular varieties of dimension $4$ ...
- 2,323
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246
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Borel-Moore homology for resolution of singularities
Let $X$ be a singular projective variety. Denote by $Z$ the singular locus of $X$. Consider the resolution of singularities $$\pi: \widetilde{X} \to X$$
Denote by $E$ the exceptional divisor. We know ...
- 2,323
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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 ...
- 480
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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, ...
- 345
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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 ...
- 305
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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 ...
- 3,318
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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 $\...
- 25
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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}...
- 241
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119
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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 $k$ ...
- 282
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327
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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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150
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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 ...
- 309
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200
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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 ...
- 3,730
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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 ...
- 232
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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 ...
- 481
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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 $G \subset SU(m)$?
- 1,441
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138
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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 ...
- 337
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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 ...
- 7,722
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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 ...
- 595
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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 ...
- 609
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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 $\...
- 25
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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 ...
- 893
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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" $...
- 7,746
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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 ...
- 1,677
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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 ...
- 7,746
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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 ...
- 1,947
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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 ...
- 893
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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 ...
- 295
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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 $...
- 263
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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 ...
1
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152
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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 ...
- 83
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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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