14
votes
Accepted
Variety without a compactification whose complement is smooth
Presumably, you want $\bar X$ to be smooth as well. Then there are many examples. Here is a simple one. Let $\bar Y$ be smooth projective curve of positive genus. Now remove at least two points to get ...
- 35.2k
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 ...
- 2,427
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 ...
- 35.2k
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 ...
- 21.3k
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 ...
- 2,300
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 ...
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. ...
- 3,998
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 ...
- 6,253
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 ...
- 42.9k
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.
- 7,300
7
votes
Lindelöf paper on meromorphic singularities
It’s available at Biodiversity Library:
https://www.biodiversitylibrary.org/item/49762#page/413/mode/1up
- 31.5k
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
...
- 79.9k
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_{\...
- 657
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 ...
- 42.9k
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 ...
- 29.1k
6
votes
Can Coulomb branches have symplectic resolutions?
In this paper, it was proved that Coulomb branches of quiver gauge theories of type ADE are (generalized) affine Grassmannian slices. There is an example where it is isomorphic to the minimal ...
- 2,951
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 ...
- 7,300
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 ...
- 20.5k
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 $T$-...
- 27.8k
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 ...
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) ...
- 1,073
5
votes
Accepted
Resolution of Gorenstein rational singularities on a surface
The explanation given in the text indeed seems to be too terse. What follows is taken from an insert I have in my copy of that book (since the argument I came up with when I read the article many ...
5
votes
Cohomology of resolution of singularity
Given a resolution $\pi:\tilde X \to X$, you can ask whether the pullback morphism $\pi^*:H^k(X) \to H^k(\tilde X)$ is injective for some (or all) $k$. As Donu points out, the mixed Hodge structure ...
- 180
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 ...
- 28.7k
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 ...
- 20.5k
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 $\tilde{O}$ ...
- 39.3k
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 ...
- 3,625
5
votes
Accepted
Borel-Moore homology for resolution of singularities
Borel-Moore homology is actually the easiest variant of (co)homology to prove this in. More generally let $\pi \colon Y \to X$ be a proper morphism which induces an isomorphism away from a closed ...
- 915
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.
- 28.7k
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