# Tag Info

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### 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  and . In particular, any weighted projective space $\mathbb{P}(\mathcal Q)$ is irreducible,...
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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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### 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 ...

### 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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### 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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### 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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### 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 ...

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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### When is a singular point of a variety ($\mathcal{C}^\infty$-) smooth?

As Francesco Polizzi said, the answer is in fact no. Here, a more general answer. Let $X\subset \mathbb{C}^n$ be a complex analytic set with $d=\dim X$ and $x\in X$ be a singular point, then: 1) ...

### 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 ...

### Can the constant rank theorem for smooth manifolds be generalized to nonconstant rank?

Your motivating question has a negative answer: Consider the inclusion $\iota:S^m\to\mathbb{R}^{m+1}$, which has rank at most $m$. If there were a smooth extension $f:D^{m+1}\to\mathbb{R}^{m+1}$ of ...
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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 ...

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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### 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, ...
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 ...