15
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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 [1] and [2].
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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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\...
14
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 ...
- 503
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 ...
- 2,352
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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}\...
- 2,729
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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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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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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 ...
- 371
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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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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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 (...
- 9,959
9
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What is the ideal of hypersurfaces singular at a given irreducible variety?
If $X=\mathbb{V}(I)$ is given by the ideal $I$, then the $m$th symbolic power $I^{[m]}$ consists of all those functions vanishing to multiplicity $m$ at the generic point of $X$. Thus a hypersurface $\...
- 1,256
8
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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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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 ...
8
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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
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What does the Jacobian of a vector field at an equilibrium tell you about local behavior of integral curves when the Jacobian is not a stable?
The Jacobian alone doesn't have the information you need. For example, consider the two vector fields
$$f(x, y) = (-x^2, -y)$$
and
$$g(x, y) = (-x^3, -y)$$.
They have an isolated equilibrium at the ...
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7
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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 $f = (x^2+y^...
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When is a singular point of a variety ($\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) ...
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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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votes
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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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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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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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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7
votes
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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
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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.
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