Questions tagged [resolution-of-singularities]
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245 questions
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canonical divisor on singular curves with nodal point
What's the definition of canonical divisor(or whatever related concept) on singular curve with nodal point. More generally, what the definition of canonical divisor on a singular variety X, which is ...
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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 ...
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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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Resolution of rational surfaces
Let $S$ be a rational singular complete algebraic surface over $\mathbb{C}$. Let $\phi:\tilde{S}\to S$ be a resolution of singularities with minimal possible Picard rank (i.e. minimal $\mathrm{dim}(...
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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,...
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Toric Fan for the Du Val's singularities D_n and E_n
Let us consider the Du Val's singularities.
i.e. https://en.wikipedia.org/wiki/Du_Val_singularity.
It is well known that they are classified by ADE, because the exceptional divisors arising in the ...
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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 ...
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Igusa zeta functions of univariate polynomials: $\mathbb{Z}_p$ or $\mathbb{Q}_p$ in this statement
Let $f\in\mathbb{Z}_p[X]$ and let $Z_{f,p}(T)\in\mathbb{Z}_{(p)}(T)$ be the $p$-adic Igusa zeta polynomial (i.e. $Z_{f,p}(p^{-s})$ is the $p$-adic Igusa zeta function in the complex variable $s$, with ...
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On mixed $p$-adic Hodge theory
Does mixed $p$-adic Hodge theory exist? Can we extend the scope of comparison theorems using simplicial resolutions a la Deligne? Do we get 3 opposite filtrations as in classical mixed Hodge theory, ...
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Intuition behind RDP (Rational Double Points)
Let $S$ be a surface (so a $2$-dimensional proper $k$-scheme) and $s$ a singular point which is a rational double point.
One common characterisation of a RDP is that under sufficient conditions there ...
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Bijective restriction of the normalization morphism
Let $X$ be an integral separated scheme of finite type over $\mathbb{C}$. Consider the normalization morphism $f:X'\rightarrow X$. Can we always find an affine open $U\subset X'$ such that $f|_U:U\...
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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 $\...
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Blow up of 9 points in 3-fold and intersection of strict transforms
Suppose we have blown up a variety $X$ at some points $P_j$ so that we introduce exceptional divisors $E_j$ in $\widetilde X$; what is the general strategy to determine the intersections of these ...
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Automorphisms of singular hypersurfaces
Let $X\subset\mathbb{P}^{n+1}$ be an irreducible and reduced hypersurface of degree $d$.
A theorem by Matsumura and Monski asserts that if $n\geq 2$, $d\geq 3$, $(n,d)\neq (2,4)$ and $X$ is smooth ...
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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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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 ...
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Some naive questions on crepant resolutions of singularities
I have some clarifying questions about crepant resolutions of singularities. The definition that I am aware of is that if $f:\tilde X \to X$ is a resolution of singularities, then the resolution is ...
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Singularities of rational quartic surfaces
Let $X\subset \mathbb{P}^3$ be an irreducible quartic surface, defined over an algebraically closed field $k$. Suppose that $X$ is rational (i.e. birational to $\mathbb{P}^2$). Is is true that $X$ has ...
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Bertini-type theorem for strict transform
Let $(X,o)$ be an isolated, normal singularity of dimension at least $3$. Let $\pi: \widetilde{X} \to X$ be a resolution of singularity of $X$. Is it true that for a general hypersurface $H \subset X$ ...
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On Remmerts reduction
Let $(X,0)$ be a normal surface singularity. An let $\pi: \tilde{X} \to X$ be the minimal resolution. Now, we can apply a result of Oliveira (exploiting previous work by Laufer) and obtain a 1-...
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Examples of explicit computations of log-resolutions
I have been working with log-resolutions lately and learning more about them. I am aware that in general producing explicit log-resolutions is difficult, but I was wondering if this has been done in ...
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Comparisons of log canonical thresholds
Premise
Let $K$ be a field of characteristic zero and $f\in K[X_1,\dots,X_m]$. By Hironaka's theorem, there exists a log resolution (over $K$) of the ideal $(f)$. Let $\{(N_i,\nu_i)\}_i$ be the ...
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Is canonical model always with canonical singularity
Let X be a smooth variety, take the proj of canonical ring of X and denote it by Y. Is Y always a canonical variety? I know it's true for general type variety. Thank in advance.
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When does a discrepant toric resolution induce a crepant resolution of a subvariety?
Let $Y$ be a complete intersection in a complete simplicial toric variety $X_\Sigma$ such that $\DeclareMathOperator{Sing}{Sing}\Sing(Y)\subset\Sing(X_\Sigma)$. Suppose that $\phi:X_{\widehat{\Sigma}}\...
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Do arithmetic schemes have non-singular alterations?
Let $X$ be an integral normal flat finite type scheme over $\mathbb{Z}$.
Does there exist a proper surjective generically finite morphism of schemes $Y\to X$ with $Y$ an integral regular ...
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Can nonflat deformations of singularities always produce Cohen-Macaulay rings?
To make the question in the title precise, let me phrase it like this. Consider a complete local ring
$$ A := \mathbb{C}[[x_1, \dotsc, x_n]]/(f_1, \dotsc, f_m) $$
and, for definiteness, assume that $...
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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 ...
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Is dimension invariant under blow-ups?
Let $X'\rightarrow X$ be a blow-up of a finitely dimensional scheme $X$ in a center $D$.
Under which assumptions one has $\dim X'=\dim X$? Do you know a proof or a reference for a proof? Do you know ...
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Hodge numbers of a Calabi-Yau 3-fold via deformation theory
In their paper "Calabi-Yau 3-folds and Moduli of Abelian Surfaces I", Gross and Popescu calculate the hodge numbers of smooth CY 3-folds obtained in the following way (see Remark 4.11 of their paper): ...
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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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Skyscraper sheaf on a stack associated to a singular surface
Suppose $X$ is a normal projective surface with a du Val singularity. In this case, we know a crepant resolution $Y$ exists, and results of Kawamata (https://arxiv.org/abs/0804.3150, Corollary 3.5) ...
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resolution for the du Val's $(A_3)$-singularity
For the $A_m$-singularity, it can be viewed as the singular part of $\mathbb{C}^2/\mathbb{Z}_m$. The action of $\mathbb{Z}_m$ on $\mathbb{C}^2$ is defined as following
$$
\bar{1} \cdot (z,w) = (z e^{\...
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Resolution of Gorenstein rational singularities on a surface
I am reading Artin's notes "Lipman's Proof of Resolution of Singularities for Surfaces" from the book "Arithmetic Geometry". I am very confused by the proof of Lemma $6.5.$ (I am formulating it below ...
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Cohomology of a structure sheaf of a normal affine variety
I can't find the reference for the following fact:
Let $X$ be an affine variety and let $Y$ be its smooth resolution. $H^0(X,\mathcal{O}_x)=H^0(Y,\mathcal{O}_Y)$ if and only if $X$ is normal.
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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 ...
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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 ...
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Hitchin fibration and Springer resolution
Let C be a curve and let us assume $G=GL_N$ and $\mathfrak{g}=\mathfrak{gl}_N$ for simplicity. The moduli space $\mathcal{M}_H(C,G)$ of $G$-Higgs bundles admits the Hitchin fibration $\pi: \mathcal{M}...
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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 ...
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Minimal non-klt center of asymptotic linear system
Let $(X,\Delta)$ be a klt pair and $D $ a $Q $-Cartier divisor on $X $ such that the ring of sections of $D $ is finitely generated. Let $c$ be the log canonical threshold of the asymptotic linear ...
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Why is resolution of singularities useful/important?
Why is it important/useful to resolve singularities generally, and in particular in algebraic geometry? I understand that one gets a nicer geometry, but how does it help one understand the geometry of ...
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example of quintics with 5 ordinary triple point
I know we can bound the triple point on quintics in cp^3 by 5. But how to write down quintics with 5 ordinary triple point (here are simple elliptic singularity)explicitly?
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Do the cohomology groups of the structure sheaf of a smooth resolution depend on the resolution?
Let $X$ be an affine variety. Let $Y$ be smooth and let the map $f\colon Y\rightarrow X$ be proper birational. We will call $Y$ a smooth resolution of $X$.
Do the cohomology groups $H^i(Y,\mathcal{O}...
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Resolving $\mathbb Z_n$ action on $\mathbb C^2$
Consider a diagonal action of $\mathbb Z_n$ on $\mathbb C^2$ generated by $(z_1,z_2)\to (\mu^pz_1,\mu^qz_2)$, with $\mu^n=1$.
Question. Is it always possible to find a smooth blow up $X\to \mathbb ...
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surjectivity of double dual map for weil divisors on normal varieties
If $X$ is a normal complex variety, and $D$ is an effective $\mathbb{Q}$-Cartier Weil divisor, then there is a natural map $\mathcal{O}(D) \otimes \mathcal{O}(-D) \rightarrow \mathcal{O}_X$. My ...
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smoothing of isolated surface singularity
I want to know when an isolated surface singularity can be smoothed, especially for log canonical isolated surface singularity. Is there any good reference. Thanks in advance.
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simple elliptic surface singularity
Suppose X is a one dimension torus and L is a line bundle over X, I think one class of log canonical surface singularity comes from the contraction of one elliptic curve from the total space L. My ...
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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 ...
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Singularities of fibrations 2
This question is related to my previous question:
Singularities of fibrations
Assume that $X$ is a complete intersection irreducible $3$-fold in a product of projective spaces. So that $X$ is ...
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