# Tag Info

### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### 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}...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $T$-...
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### 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 ...
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### 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 ...
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### 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) ...
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### 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 ...
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### 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 ...
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### 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 $\tilde{O}$ ...
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