# Tag Info

### 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,...
• 62.3k
Accepted

### 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 ...
• 19.2k
Accepted

### 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 .
• 4,011
Accepted

• 2,699
Accepted

### 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 ...
• 423
Accepted

### 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

### 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 ...
• 310
Accepted

• 98.5k
Accepted

### 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 ...
• 2,312
Accepted

### 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 ...
• 1,104

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 (...
• 8,884

### 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 ...
• 15.5k
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 ...
Accepted

### 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 ...
• 62.3k
Accepted

### 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 ...
• 78.6k

### 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 $$\operatorname{Pic}(\mathbb{P}^n) \to \operatorname{Pic}(X)$$ ...
• 62.3k
Accepted

### 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 ...
• 34.7k

### 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 ...
• 37.5k
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 ...