# Questions tagged [resolution-of-singularities]

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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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### Birational model of a log smooth pair

Given a log smooth pair $(X,B)$ with a reduced boundary divisor $B$, consider a birational model $\pi:X' \to X$ and a boundary divisor $B'$ which is given by $K_{X'}+B'=\pi^*(K_X+B)$. Here is my ...
The following definitions are from lecture notes of Némethi. A surface singularity $(X,0)$ is defined by $$(X,0) = (\{ f_1 = \ldots = f_m=0 \}) \subset \mathbb (\mathbb{C}^n,0),$$ where $f_i : (\... 0answers 236 views ### Artin's “On Isolated Rational Singularities of Surfaces” My question refers to M. Artin's paper "On Isolated Rational Singularities of Surfaces"; more precisely the proof of Thm 4 on page 133. Here the relevant excerpt: The Setting: Let$\bar{V}=Spec(A)$... 0answers 95 views ### Are there any explicit (prime-to-l) alterations for interesting varieties (or schemes)? I have read that it is easier to find regular alterations of varieties than their resolutions of singularities (moreover, I believed in this sentence when I read it). My question is: do there exist ... 1answer 110 views ### Singular locus of a linear system of hyperplane sections Let$X\subset\mathbb{P}^N$be a rational smooth projective irreducible non degenerated variety of dimension$n=\dim(X)$and let $$\mathcal{H}=|\mathcal{O}_X(1) \otimes\mathcal{I}_{{p_1}^2,\dots,{p_l}^... 0answers 145 views ### 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 ... 1answer 129 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 ... 0answers 103 views ### 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? 0answers 54 views ### 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 ... 0answers 109 views ### 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}(... 1answer 103 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,... 0answers 118 views ### 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 ... 1answer 139 views ### 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 ... 0answers 225 views ### 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 ... 0answers 711 views ### 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, ... 0answers 54 views ### 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\... 1answer 151 views ### 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 \... 0answers 63 views ### 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 ... 1answer 78 views ### Image of a quiver variety under natural morphism We know that the natural morphism \pi:\mathfrak{M}_{\theta}(Q,\mathbf{v},\mathbf{w})\rightarrow \mathfrak{M}_0(Q,\mathbf{v},\mathbf{w}) between a smooth and affine quiver variety is not necessarily ... 1answer 192 views ### 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 ... 0answers 115 views ### 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? 1answer 175 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 ... 1answer 220 views ### 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 ... 0answers 105 views ### 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 ... 0answers 230 views ### 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 ... 0answers 36 views ### 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-... 0answers 81 views ### 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 ... 1answer 293 views ### 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. 1answer 235 views ### 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 ... 0answers 174 views ### Desingularization of the zero section of TM as the manifold of singularities of the geodesic flow However the method of "Blowing up of singularities" is initially introduced for an isolated singularity, but this method have been generalized to blowing up of a "Manifold of singularities"... 1answer 83 views ### 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 ... 0answers 154 views ### 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) $$and, for definiteness, assume that ... 1answer 246 views ### Comparisons of log canonical thresholds Premise Let K be a field of characteristic zero and f\in K[X_1,\dots,X_m]. By Hironaka's theorem, there exists a log resolution (over K) of the ideal (f). Let \{(N_i,\nu_i)\}_i be the ... 1answer 297 views ### Hironaka's theorem and smooth completion Hironaka's theorem states that for any algebraic variety (analytic space) X there exists a smooth variety (complex manifold) X' and a morphism f : X' \rightarrow X such that f restricted to X ... 0answers 117 views ### Reference for certain resolution of singularities formulation I want to use the following resolution of singularity statment as found in Soule et al, Lectures on Arakelov Geometry, p. 40: Y is a separated algebraic variety of finite type over \mathbb{C}, Z... 0answers 155 views ### example of torsion of higher direct image sheaf I'm reading kollar's paper about higher direct image of dualizing sheaf. Suppose f: X-Y is morphism, X smooth,Y normal. He mentioned usually the higher direct image of structure sheaf is "bad," and ... 0answers 163 views ### Skyscraper sheaf on a stack associated to a singular surface Suppose X is a normal projective surface with a du Val singularity. In this case, we know a crepant resolution Y exists, and results of Kawamata (https://arxiv.org/abs/0804.3150, Corollary 3.5) ... 1answer 222 views ### resolution for the du Val's (A_3)-singularity For the A_m-singularity, it can be viewed as the singular part of \mathbb{C}^2/\mathbb{Z}_m. The action of \mathbb{Z}_m on \mathbb{C}^2 is defined as following$$ \bar{1} \cdot (z,w) = (z e^{\... 0answers 213 views ### 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. 1answer 440 views ### 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 ... 0answers 163 views ### 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 ... 1answer 122 views ### 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 ... 1answer 177 views ### strict transform under resolution of singularity along a singular$\mathbb{Q}$-Cartier divisor 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$E\cong \mathbb{P}(1,1,2)$are ... 2answers 323 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$\colonf \colon ...
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 ...