13
votes

### Is canonical model always with canonical singularity

I believe that $(Y,B)$ is always klt for some boundary $B$. In fact by Theorem 5.2 of https://projecteuclid.org/download/pdf_1/euclid.jdg/1090347529, after passing to a truncation of the ...

13
votes

Accepted

### Cohomology of resolution of singularity

In general, the answer is no. It already fails for a nodal curve. In fancier terms, you can understand the obstruction as follows: if $X$ has a resolution $\tilde X$, and the cohomology of $X$ injects ...

11
votes

### Perfectoid approach to resolution of singularities in char $p$

Somehow that question slipped my radar, sorry!
The truth is that shamefully I'm not able to say much, as I don't have a strong knowledge of resolution of singularities. But at least so far, the flow ...

10
votes

### Normalization of complete intersection

Consider $A=k[[x^3,x^2y,y^3]]\subset k[[x^3, x^2y, xy^2, y^3]]=B$. $B$ is the integral closure of $A$, $A$ is a hypersurface, but $B$ is not Gorenstein.

10
votes

Accepted

### Cohomology of tangent sheaf of a singular hypersurface

Put $d:=\deg(X)$. From the exact sequence
$$0\rightarrow \mathcal{O}_X(-d)\rightarrow \Omega ^1_{\mathbb{P}^n|X}\rightarrow \Omega ^1_X\rightarrow 0$$you get an exact sequence $\ 0\rightarrow T_X\...

10
votes

### Crepant resolutions of cDV singularities?

Background. The threefold compound du Val singularities have been introduced by Miles Reid in the 1980s [R1, R2, R3]. Their geometric description is that a general hyperplane section through the ...

9
votes

Accepted

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

It is easy to see that the canonical class of $X'$ is trivial (combine the formula for the canonical class of the blowup of $A^3$ and the adjunction formula). Hence $X'$ is crepant. But a crepant ...

8
votes

Accepted

### Do the cohomology groups of the structure sheaf of a smooth resolution depend on the resolution?

Edit. This follows from the Elkik-Fujita Vanishing Theorem. There is a more general vanishing theorem due to Elkik and Fujita. One version of this theorem (where I read the theorem) is Theorem 1.3.1 ...

Community wiki

7
votes

### Hitchin fibration and Springer resolution

I will try to answer the first question only.
As in the remarks, the canonical reference is
Beauville, Narasimhan, Ramanan, Spectral curves and the generalised theta divisor. J. Reine Angew. Math. ...

7
votes

Accepted

### Resolving $\mathbb Z_n$ action on $\mathbb C^2$

I think the answer is no. In your notation, take $n=5$, $p=1$ and $q=2$. If you consider the blowup at the origin $X\to \mathbb C^2$, then $\mathbb Z_5$ acts on $X$ with two isolated fixed points: at ...

7
votes

Accepted

### Some naive questions on crepant resolutions of singularities

Your definition is the usual one. More or less. Probably you should assume that $K_X$ is Cartier or at least $\mathbb Q$-Cartier for the definition to even make sense.
I don't think there is a ...

7
votes

Accepted

### Computing the invariants of ball quotient surfaces

I'm assuming that you know "where" in the commensurability class your lattice is. By this, I mean you perhaps have $\Gamma$ as a subgroup of some principal arithmetic lattice $\Lambda$ of ...

7
votes

Accepted

### Is there a "minimal" Whitney stratification of a complex hypersurface?

The answer is yes for any reduced equidimensional analytic space. This is the proposition 3.2 (and remark after the proof) page 479 of Variétés polaires II by Bernard Teissier.

7
votes

### Lindelöf paper on meromorphic singularities

It’s available at Biodiversity Library:
https://www.biodiversitylibrary.org/item/49762#page/413/mode/1up

6
votes

Accepted

### example of quintics with 5 ordinary triple point

Choose five points $P_i$ in general linear position (all choices are
equivalent under ${\rm PGL}_4$, so you might as well put four of them
at the coordinate vectors and the fifth at $(1:1:1:1)$); then
...

6
votes

Accepted

### Resolution of singularities in étale cohomology

I am turning Giulia's comment into a CW answer. At over 400 pages, I think that "Travaux de Gabber", published in Astérisque, ought to count as a "comprehensive treatment of [some aspects of ] étale ...

Community wiki

6
votes

### Equivariant resolution of singularities

To any variety $X$ over a field of characteristic zero, say $k$, one can attach a resolution of singularities, say $X'\to X$, with the following properties:
$X'\to X$ is an isomorphism over $X_{\...

6
votes

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

Just to spell out what Sasha means, here is a way to see that the resolution in crepant. Once you believe in adjuction,
$$
K_{X'} = (K_{\tilde{\mathbb{A}^3}} + X')_{\mid X'} = (\sigma^*K_{\mathbb{A}^3}...

6
votes

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

By the Hochster-Roberts Theorem it is Cohen-Macaulay
The canonical sheaf of $\mathbb A^n$ is $G$-invariant (because the elements of $G$ have det=1) and hence it descends to the canonical sheaf of the ...

6
votes

Accepted

### Singularities of PL embedding of surface in a contractible 4-manifold

One reference for these matters is Rourke-Sanderson "Introduction to piecewise-linear topology". In particular, if $S\subset M$ is a simplical submanifold, then Corollary 4.2 in this book ...

5
votes

Accepted

### Singularities of $3$-folds

It can have bad singularities and dimension doesn't matter. Take an arbitrarily singular $X$ of dimension $d$. There is always a finite generic projection $X \to Z = {\mathbb{P}}^d$ (and $Z$ is ...

5
votes

Accepted

### Canonical sheaf of affine variety

For a toric variety, such as this cone over the 3rd Veronese of $\mathbb P^1$, the complement of the open torus orbit is an anticanonical divisor (indeed, that is the one given by the unique $T$-...

5
votes

Accepted

### Resolution of an isolated cyclic quotient singularity

Edit. As Michael Entov points out, the original example (below with strikethrough) is incorrect. (Sorry!) Here is a corrected example. It is not always possible to find an equivariant resolution ...

Community wiki

5
votes

Accepted

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

You probably want to show that the quotient scheme $\mathbb{A}^n_{\mathbb{C}}/G$ is Gorenstein.
A proof can be made along the following line. Since $G \subset SL_n(\mathbb{C})$, the canonical bundle ...

5
votes

### Hitchin fibration and Springer resolution

I'll try to answer the second and third questions. My preferred way to organize this circle of ideas is to think of the following ladder of theories :
Representation theory of $\mathfrak{g}$ (or) ...

5
votes

### Minimal resolution of singularities of surfaces

The statement that $\pi^{-1}(x)$ is (geometrically) connected for all $x \in X$ is Stein factorisation or Zariski's main theorem; see e.g. [Hartshorne, Cor. III.11.5].
For the other statement, it ...

5
votes

Accepted

### Normal forms of ADE singularities

This may not be exactly what you are asking about, but what follows has some references. I'm guessing you are already aware of:
V. I. Arnolʹd, Critical points of smooth functions, and their normal ...

5
votes

Accepted

### Perfect complexes of plane nodal cubic curve

The kernel is zero.
Indeed, on the one hand, since $\pi$ is an isomorphism over the complement of $O$, any object in $\operatorname{Ker}(\pi^*)$ is supported at $O$.
On the other hand, if $\tilde{O}$ ...

5
votes

Accepted

### Question about surface singularities

(1) As Marco Golla wrote in a comment: Némethi András, On the Ozsváth-Szabó invariant of negative definite plumbed 3-manifolds, Geom. Topol. 9 (2005), 991-1042 gives examples of rational non-quotient ...

4
votes

Accepted

### Do arithmetic schemes have non-singular alterations?

This is Theorem 8.2 in de Jong's original paper [dJ].
[dJ] de Jong, A. J., Smoothness, semi-stability and alterations. Publ. Math., Inst. Hautes Étud. Sci. 83, 51-93 (1996). ZBL0916.14005.

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