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Questions tagged [resolution-of-singularities]

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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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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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Locally toric resolutions of compactifications

Suppose $U$ is a smooth, open $n$-dimensional variety over $\mathbb{C}.$ Say $X, X'$ are two proper normal-crossings compactifications of $U$. Call a map $m: X'\to X$ a modification if it is an ...
Dmitry Vaintrob's user avatar
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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 ...
George's user avatar
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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 ...
David Hubbard'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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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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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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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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Divisorial contraction to a non-normal variety

Consider a divisorial contraction $f:X\rightarrow Y$, between projective varieties, contracting an irreducible divisor $D\subset X$ to a subvariety $Z\subset Y$ of codimension at least two, and which ...
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Resolution of pairs in characteristic p

Let $R$ be a complete DVR of characteristic $p$, say $R=\mathbb{F}_p[[t]]$, and $X$ be a reduced scheme of finite type over $R$. Let also $X_s$ denote the special fiber of $X$. If I understand ...
Kostas Kartas's user avatar
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Is toroidalization local?

Let $f:X \to Y$ be a surjective morphism of smooth projective varieties, $D$ be a simple normal crossings divisor on $X$ and $U_Y \subset Y$ be an open subset over which $(X,D)$ is log smooth (in the ...
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Log canonical centers of toric (and toroidal) varieties

Q1: Let $(X,B)$ be a toric variety. There exists a toric resolution of singularities $f:(Y,E) \to (X,B)$. Here is my question: Is any lc center of $(X,B)$ an irreducible component of an intersection ...
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Terminal and log canonical singularities

Let $D$ be a divisor with at most terminal singularities in a smooth projective variety $X$. Is the pair $(X,D)$ log canonical?
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Test rational singularities after forming invariants

Let $R$ be a normal local domain of dimension $2$ and odd residue characteristic endowed with the action of the finite group $G \cong \mathbb{Z}/2\mathbb{Z}$. Suppose that the ring of invariants $R^G$ ...
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Cubic 3-fold singular along a curve

Does there exists a cubic or quartic $3$-fold $X\subset\mathbb{P}^4$ such that $Sing(X)$ is a smooth curve $C$ of genus $g(C)\geq 2$ and $X$ has $A_1$-singularities along $C$?
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Uniformity of the set of poles of Igusa local zeta functions

Let $Ω_p$ denote the set of the real parts of the poles of the Igusa zeta function of a polynomial $f∈\mathbb{Z}[X_1,…,X_m]$ (assume $f(0)=0$ so that $\Omega_p\ne \emptyset$) at the prime p. From ...
Maurizio Moreschi's user avatar
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Reference for certain resolution of singularities formulation

I want to use the following resolution of singularity statment as found in Soule et al, Lectures on Arakelov Geometry, p. 40: $Y$ is a separated algebraic variety of finite type over $\mathbb{C}$, $Z$...
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example of torsion of higher direct image sheaf

I'm reading kollar's paper about higher direct image of dualizing sheaf. Suppose f: X-Y is morphism, X smooth,Y normal. He mentioned usually the higher direct image of structure sheaf is "bad," and ...
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Local weak factorization

This is a follow-up to question Locally toric resolutions of compactifications, answered by Jason Starr. In a series of papers (see https://arxiv.org/abs/math/9904076), Jaroslaw Wlodarczyk proves ...
Dmitry Vaintrob's user avatar
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Strict transform under resolution of singularity along a singular $\mathbb{Q}$-Cartier divisor

$\DeclareMathOperator\Bl{Bl}$Let $f: Y=\Bl_0^\omega(\mathbb{C}^3)\to \mathbb{C}^3$ be a weighted blow up of $\mathbb{C}^3$ with weights $w(x,y,z)=(1,1,2)$. Then $Y$ and the exceptional divisor $E\cong ...
Mingyi Zhang's user avatar
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log canonical surface singularity

For a log cannonical surface singularity, I guess there is classification about the configuration of the exceptional divisor of its minimal resolution. I wonder if there is some specific example or ...
xin fu's user avatar
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Total space of canonical bundle on toric del pezzo surface

Let $X$ be a toric del pezzo surface with a full cyclic strong exceptional collection of line bundles, say $\mathbb{E}:=(E_1,\ldots,E_n)$, consider the total space of anti-canonical bundle $\omega_X^*$...
George Baobei's user avatar
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(Strict) canonical singularity with no crepant resolution

We know if a normal singular $\mathbb{Q}$-factorial variety $X$ has a crepant resolution, then it is canonical but not terminal. What can we say about the converse? To be more precise: Is there a ...
Xuqiang QIN's user avatar
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271 views

Desingularization of subvariety

Let $X$ be a smooth projective complex variety. Let $Y \subset X$ is a singular closed subvariety of $X$. Does there exist a birational morphism $\pi: \widetilde{X} \to X$, such that the proper ...
Edward Teach's user avatar
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116 views

Constructive Resolution of Toric Singularities via Model Theory

Do there exists some language $\mathcal{L}$ of rational polyhedral cones in rational vector spaces and a theory $T$ over $\mathcal{L}$ whose models $\mathcal{M}$ are resolutions of toric singularities?...
Joaquín Moraga's user avatar
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191 views

Resolution of indeterminacies for a map to Grassmannian of planes

Let $X\subset \mathbb P_k^N$ be a $n$-dimensional smooth projective variety ($n\geq 2$) and $\phi_l:X^l\dashrightarrow Gr(l,N+1)$ ($l\leq n+1$) be the natural rational map which associates to a ...
user3001's user avatar
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Global minimal model over a non-affine base

Remark 10.1.8 in Liu's book (AG and Arithmetic curves) says that over a non-affine base (base is always assumed to be a Dedekind scheme of dim 1), the minimal regular model of a (smooth projective) ...
QcH's user avatar
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compatible resolutions of singularities

Let $U$ and $V$ be complex vector spaces with an action of a finite group $G$. Denote by $P_{G,d}$ the space of $G$-equivariant polynomial maps $U\longrightarrow V$ with degree less than or equal to $...
Brett Parker's user avatar
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2 answers
508 views

Base change of a finite morphism

Let $X$, $Y$, $Z$ be integral schemes of finite type over a field $K$ (i.e., locally affine opens are finitely-generated algebras over $K$). Suppose we have the following condition$\colon$ $f \colon ...
Pierre's user avatar
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Hironaka's theorem and smooth completion

Hironaka's theorem states that for any algebraic variety (analytic space) $X$ there exists a smooth variety (complex manifold) $X'$ and a morphism $f : X' \rightarrow X$ such that $f$ restricted to $X ...
Federico Barbacovi's user avatar
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273 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
216 views

Pushforward of locally free sheaves and resolution of singularities

Let $X$ be an noetherian, affine, isolated of dimension at least $2$ and $\pi:\widetilde{X} \to X$ a resolution of singularities. Let $\mathcal{E}$ be a locally-free sheaf on $\widetilde{X}$ such that ...
Ron's user avatar
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1 vote
1 answer
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Infinitesimal deformation of strict transform

Let $X$ be an affine, complex surface with isolated singularities and $\pi:\widetilde{X} \to X$ be a resolution of singularities (not necessarily minimal) i.e., $\widetilde{X}$ is non-singular and $\...
Ron's user avatar
  • 2,126
1 vote
1 answer
156 views

Pushforward of $K_X+D$ on the non-snc locus

Let $f:Y\rightarrow X$ be a birational morphism of smooth projective varieties, $F$ an effective divisor on $X$, $D=f^{-1}F_{\mathrm{red}}+\mathrm{Ex}(f)$, $B$ a smooth subvariety of $Y$ contained in ...
ormula's user avatar
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1 answer
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'Reference' request: Program to work with cyclic quotient singularities.

I'm looking for program code to deal with cyclic quotient singularities on normal surfaces. In particular, at the moment I need that given a singularity $p$ like $p=\frac{1}{n}(1,a)$ the code computes:...
Jesus Martinez Garcia's user avatar
1 vote
1 answer
151 views

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 $4$ ...
user45397's user avatar
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1 answer
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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
1 vote
1 answer
383 views

Construction of log canonical singularity

I know there's classification about normal log canonical surface singularity in the sense of configuration of exceptional curves. There is one type of log canonical singularity(not klt) whose ...
xin fu's user avatar
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1 answer
471 views

Serre's conditions under blow-ups, Blowup and normalization

Suppose $X = \mathbb{Z}[x, y, z]/(f,g)$ is a 2-dimensional Cohen-Macaulay surface. In particular, $X$ satisfies Serre's condition $S_2$. Suppose it is irreducible, reduced but not normal. $\bf{...
LMN's user avatar
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1 vote
1 answer
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Importance of Denjoy-Carleman classes as a class.

Denjoy-Carleman classes of differentiable functions, say in Roumieu's form: Given a log-convex sequence $M_n$ of positive number denote by $C_M=C_M(\mathbb{R}^n,0)$ the ring of germs of $C^\infty(\...
O.R.'s user avatar
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1 answer
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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 $...
Yuanjiu Lyu's user avatar
1 vote
1 answer
209 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
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1 vote
1 answer
160 views

Generators of a graded algebra defining bundle over elliptic curve

I have a question about a statement from P. Wagreich's paper "Elliptic Singularities of Surfaces" (page 425): We consider an elliptic curve $X$ and a line bundle (=invertible sheaf) $L$ on $X$. Then,...
user267839's user avatar
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1 vote
1 answer
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Singular irreducible quadrics

Let $Q\subset\mathbb{P}^n$ be the quadric hypersurface defined by $$x_0^2+x_1^2+...+x_k^2 =0.$$ If $2\leq k\leq n-1$ then $Q$ is irreducible and $Sing(Q)$ is a linear space of dimension $n-k-1$. If $...
user avatar
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0 answers
92 views

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 ...
Richard's user avatar
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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 $E_j$ ...
George's user avatar
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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 ...
Ji Woong Park's user avatar
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62 views

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 ...
George's user avatar
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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 ...
user341068576's user avatar