15 votes

Which weighted projective spaces (and their finite quotients) are local complete intersections?

Regarding your question about weighted projective spaces, a lot is known about them, see for instance [1] and [2]. In particular, any weighted projective space $\mathbb{P}(\mathcal Q)$ is irreducible,...
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14 votes
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Obtaining non-normal varieties by pushout

A number of people have asked me for a reference since I wrote down that answer in the other question so I'll try to write a reference here (I'm sure some experts knew it before though). I originally ...
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  • 19.2k
14 votes
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A geometric characterization of smooth points of a complex algebraic variety

The answer to all three of your questions is yes.See the book by E M Chirka titled Complex Analytic Sets pages 189,190 and 120 .These questions are local so this is true on Kahler manifolds .
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14 votes
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The homology groups of the smooth locus of a singular variety

I am adding some additional details to the comment above, since somebody else asked me about this recently. Results about extensions of cohomology classes to all of $X$ from an open subset $U=X\...
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Does a resolution of a rational singularity have rationally connected fibers?

No. For instance the cone over an Enriques surface (with respect to any projective embedding) has rational singularity, but Enriques surface is not rationally connected.
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  • 32.1k
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 ...
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  • 2,312
13 votes
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Fixed point scheme of finite group Cohen-Macaulay?

Here is a simpler example than the one I left before, using the same strategy. Let $$X = \{ x_1 x_3 = x_1 x_4 = x_1 x_5 = x_2 x_4 = x_2 x_5 = x_3 x_5 = 0 \} \subset \mathbb{C}^5.$$ This is the reduced ...
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13 votes

Are Du Val singularities smoothable?

Du Val singularities are hypersurface singularities, hence they can be smoothed --- just replace the defining equation $F(x,y,z) = 0$ by the equation $F(x,y,z) = \epsilon$.
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  • 32.1k
12 votes

Link of a singularity

More generally, consider the singularity given by $$x_1^2+\cdots+x_{n+1}^2=0$$ in $\mathbf{C}^{n+1}$. (Your case is $n=2$ after a change of variables.) Identifying $\mathbf{C}^{n+1}=\mathbf{R}^{n+1}\...
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  • 2,699
12 votes
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Derived Category of the derived critical locus, is it the category of Matrix Factorizations?

These are indeed related. The first thing to know is that they both ``live'' (i.e., sheafify) over the critical locus (this is not saying much if you assume $W$ has isolated critical points, but ...
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11 votes
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Canonical scheme structure on the singular locus of a variety

The answer is yes. Let $X$ be a scheme of finite type over the field $k$, of pure dimension $r$; then $S_X$ can be defined as the closed subscheme of $X$ defined by the $r$th Fitting ideal of the ...
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  • 371
10 votes

Giant Rat of Sumatra singularity

I'd just like to add something about the plot of the graph of the function $z=\mathrm{f}(x,y)$. The term "singularity", in this context, does not refer to a function whose graph is singular. The ...
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10 votes
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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\...
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10 votes
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Normal form of functions $(x^2+y^2)^n+$ higher terms

The expression $(x^2+y^2)^2 + x^5 + y^5$ cannot be written in the form $(z^2+w^2)^2$ for any smooth functions $z$ and $w$ of $x$ and $y$. (Just look at the Taylor series expansion.) Similarly, $n>...
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9 votes
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A paradox on the deformation of singularities

I don't think it is true that $\mathcal X$ is $\mathbb Q$-Gorenstein. Suppose in fact that $\dim \mathcal X _t=2$ for all $t\in C$ and $\mathcal X \to Z$ is a flipping contraction with exceptional ...
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9 votes
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Cohomology of real analytic coherent sheaves

In a smooth case, the reference is Proposition 2.3 in Atiyah and Hirzebruch's Analytic cycles on complex manifolds. For a non-smooth case, I don't know the general reference, but Theoreme 3 in Henri ...
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9 votes

Link of a singularity

For singularities of the form $g(x,y)+z^n = 0$ there is a nice description: if you project onto the $xy$-plane (and you take a very small neighbourhood of the origin), you can view the link of the (...
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  • 8,884
8 votes

Can you prove Givental's conjecture on wavefronts and the icosahedron?

In http://www.sciencedirect.com/science/article/pii/S0167278998900057 (Remarks on quasicrystallic symmetries) Arnold so describes the idea of Shcherbak's proof: Now the proof of this theorem ...
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8 votes
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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 ...
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Are Du Val singularities smoothable?

Du Val singularities are indeed smoothable and, in fact, more is true: they are of class $T$, namely, they are quotient singularities admitting a $1$-parameter $\mathbb{Q}$-Gorenstein smoothing. See ...
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8 votes
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Is there a, in depth, classification of branch points in complex analysis?

Yes, there is a classification. An isolated branch point can be algebraic or logarithmic. If the branch point is at 0, algebraic means that $f(z^n)$ has a pole or removable singularity at 0. It can ...
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8 votes

Smooth complete intersections

If $X \subset \mathbb{P}^n$ is a non-degenerate, smooth complete intersection variety of dimension at least $3$, then the restriction map $$\operatorname{Pic}(\mathbb{P}^n) \to \operatorname{Pic}(X)$$ ...
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7 votes
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Castelnuovo's rationality criterion on singular surfaces?

It does not hold in general: a cone over a smooth plane cubic satisfies $q=P_2=0$ but is not rational. On the other hand if $S$ has canonical singularities and $\tilde{S} \rightarrow S$ is any ...
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7 votes

smooth quotient out of a singular variety?

Here's an example: Take $\mathbb C^2$ minus the origin and identify the points $(1,1)$ and $(-1,-1)$ and let this be $X$. Then $G=\mathbb Z/2$ acts by $(x,y) \mapsto (-x,-y)$ and the quotient is ...
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  • 37.5k
7 votes

Analytical formula for topological degree

First, let me apologize for the confusion regarding the pre-factor. It should read $(-1)^{k}(2\pi i)^{-2k-1}$. This can be seen in several ways. One way is to compare the constant $c_k$ appearing in ...
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7 votes
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Complexifying a real-analytic singularity

As David Speyer has already suggested, we have $Q_f^\mathbb{C}\simeq Q_f\otimes\mathbb{C}$, so that answers the first question (and the second question). For the third question, consider $f = (x^2+y^...
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7 votes
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Asymptotic behavior of the ratio between the largest two singular values of product of i.i.d. random complex matrices

So this is correct. The theorem that you need is the multiplicative ergodic theorem. Expressing it in your language, it states that $\frac 1n\log s_i(A_n)\to\lambda_i$, where $s_i$ is the $i$th ...
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  • 21.6k
7 votes

Link of a singularity

To add to the excellent answers already provided, here are some general facts in the case of rational surface singularities (1 and 2) and hypersurface singularities (3). Many interesting ...
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  • 6,230
7 votes
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General conditions for normality of blow-up

Let $X=Spec(R)$. Blowing-up $Z=V(I)$ is the same as to look at $Proj$ of the graded ring $R[It]=\oplus_{j\geqslant 0} I^jt^j\subset R[t]$, the Rees ring associated to $I$. Assume $R$ is a domain, ...
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