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Questions tagged [resolution-of-singularities]

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1answer
119 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 ...
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1answer
139 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 ...
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0answers
97 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 $H \subset X$ ...
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170 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 ...
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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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63 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 ...
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1answer
213 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.
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1answer
212 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 ...
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168 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"...
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50 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 ...
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151 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 $...
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1answer
205 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 ...
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1answer
180 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 ...
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0answers
89 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 $\mathbb{C}$, $Z$...
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0answers
117 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 ...
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110 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) ...
2
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1answer
166 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^{\...
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179 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.
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1answer
354 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 ...
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0answers
115 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 ...
2
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1answer
108 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 ...
2
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1answer
122 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 ...
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2answers
295 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 ...
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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 ...
2
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1answer
550 views

On the coherence of a Néron-ring

Let $A:= \underset{\lambda \in \Lambda}{\varinjlim} \,A_{\lambda}$ be an inductive limit of geometric regular local ring $(A_{\lambda}, {\frak m}_{\lambda})$, whose transition map $\phi_{\mu\lambda} \...
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299 views

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 ...
3
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1answer
201 views

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?
8
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1answer
540 views

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}...
9
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2answers
594 views

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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1answer
77 views

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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1answer
155 views

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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0answers
175 views

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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0answers
195 views

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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Hironaka's proof of resolution of singularities in positive characteristics

Recent publication of Hironaka seems to provoke extended discussions, like Atiyah's proof of almost complex structure of $S^6$ earlier... Unlike Atiyah's paper, Hironaka's paper does not have a ...
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197 views

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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0answers
147 views

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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0answers
65 views

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 ...
5
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1answer
150 views

Singularities of fibrations

Let $f:X\rightarrow \mathbb{P}^2$ be a fibration, here $X$ is a projective variety of dimension three. Assume that there exixts a smooth curve $C\subset\mathbb{P}^2$ such that for any $p\in\mathbb{P}...
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1answer
121 views

Singularities of $3$-folds

Let $X,Y,Z$ be projective $3$-folds. Assume that $Y$ is smooth and $Z$ is smooth and Fano. Moreover, assume that there is a generically finite morphism $f:Y\rightarrow Z$ admitting a factorization $f=...
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138 views

Resolving analytic normal crossings singularities

Let $X$ be a non-singular (complex) variety and $Y \subset X$ be a (reduced) irreducible subvariety with only normal crossings singularity (locally, in the analytic topology, the singularity is ...
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0answers
94 views

Resolving complete intersections of quadrics with singularities

Suppose that $X$ is a complete intersection of quadrics in $\mathbb P^n_{\mathbb C}$. Is there some straightforward procedure to resolve the singularities of $X$? For example, can one stratify ...
2
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0answers
111 views

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 $\omega_X^*$...
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0answers
61 views

Simple question about surface singularities

Given $\epsilon \in (0,1)$, is it possible to find two finite familes $\mathcal{F}$ and $\mathcal{P}$ of weighted graphs, such that the weighted graph of the minimum resolution of any $\epsilon$-klt ...
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164 views

How to compactify $\mathbb{Z}_p[x, y, z]/(xyz - p)$?

The affine scheme $U := \mathrm{Spec}(\mathbb{Z}_p[x_0, \ldots, x_d]/(x_0\cdots x_d - p))$ is regular and is a basic example of a semistable scheme over $\mathbb{Z}_p$. How does one build a proper ...
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1answer
309 views

Canonical sheaf of affine variety

Let $A=\mathbb{C}[u,x,y,w]/(uy-x^2,xw-y^2,uw-xy)$, $X=Spec A$. $A$ is a Veronese subring and from the answer of Is there a simple method to test a local ring to be Cohen Macaulay?, we can see that $X$ ...
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1answer
400 views

Resolution of an isolated cyclic quotient singularity

I am looking for a reference to the following fact which seems to be true and which is probably well-known (at least to experts in resolution of singularities): Consider an isolated cyclic quotient ...
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0answers
176 views

(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 ...
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1answer
90 views

Finitely many subvarieties as divisor

Let $X$ be a smooth projective variety over an algebraically closed field of characteristic $0$ and of dimension $n\geq 10$. Let $(C_i)_{1\leq i\leq N}$ (resp. $(S_i)_{1\leq i\leq N}$) be smooth ...
3
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2answers
212 views

Quotient of affine space by finite subgroup of SL(V) is Gorenstein

I am looking for a proof of the following fact: If $G$ is a finite subgroup of $SL_n(\mathbb{C})$ acting on $\mathbb{A}_{\mathbb{C}}^n$, then the resulting quotient scheme is Gorenstein. Thanks.
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2answers
347 views

Crepant resolution of $Y=k[x,y,z]/(xz-y^3)$

Consider the action of $\mathbb{Z}_3\subset SL_2(k)$ on $\mathbb{A}^2$, we have the quotient $Y$ as in the title. According to the classification of Du Val singularity, we know that the crepant ...