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 ...

- 2,332

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 ...

- 32.3k

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 ...

- 16.1k

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.

- 5,712

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\...

- 35.3k

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 ...

- 33.6k

9
votes

### Cone over the Veronese surface

The answers are the following.
(1) It is well known that the singularity at the vertex of the cone over the Veronese surface is isomorphic to a quotient singularity of type $\frac{1}{2}(1, \, 1, \,1)$...

- 63.5k

9
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 ...

- 2,240

9
votes

Accepted

### Cone over the Veronese surface

Let me start being a little nitpicking with the formulation of the question. The fact that $X$ is $\mathbb Q$-factorial does not in itself imply that such $a$ and $b$ exists. One also needs the fact ...

- 41.5k

9
votes

### How singular can the Stein factorization of a proper map between smooth varieties be?

The only restriction I see is that $\hat{X}$ must be normal (because $X$ is): if $\phi$ is a rational function on (some affine open subscheme of) $\hat{X}$ which is integral over $\mathscr{O}_\hat{X}$,...

8
votes

### How singular can the Stein factorization of a proper map between smooth varieties be?

$\hat{X}$ can be as bad as you want. For example, take your favorite non-Gorenstein variety $\hat{X}$ in $\mathbb{A}^N$. By Noether Lemma there is a finite morphism $\hat{X} \to \mathbb{A}^n =: Y$. ...

- 33.6k

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. ...

- 3,763

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 ...

- 6,183

7
votes

Accepted

### Blowing-up a point in the singular locus

Note that without any assumption on the singularities the answer to your question is yes. For instance, consider the hypersurface $Y = \{x_0^2+x_1^3+x_2^4\}\subset\mathbb{P}^3$. Then $Sing(Y) = [0:0:0:...

- 6,730

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 ...

- 41.5k

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 ...

- 591

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.

- 6,621

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

### How singular can the Stein factorization of a proper map between smooth varieties be?

For what it's worth, one can say the following sort of thing.
Since $Y$ is log terminal so is $(\hat{X}, -\mathrm{Ram})$. This doesn't mean much since in the pair, the boundary has a negative ...

- 19.6k

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_{\...

- 617

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 ...

- 41.5k

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
...

- 74k

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 ...

- 27.4k

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 ...

- 6,621

5
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}...

- 1,623

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 ...

- 19.6k

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) ...

- 1,033

4
votes

Accepted

### Singularities of fibrations

This is true. Actually, even better, these singularities will be terminal and Gorenstein, so as mild as it can get. Well, at least if we assume that you are working over an algebraically closed field, ...

- 41.5k

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