48

NB: This answer is directed to the questions about the real case, not the complex case, which was already treated by Francesco. In some sense, the reason you are running into these 'problems' is that you are working with the ring of real polynomials rather than the ring of real-valued analytic functions, which is also a UFD. For example, it is not hard to ...


43

The answer is in fact no. A complex variety $X$ can never be a differentiable manifold (not even of class $C^1$) throughout a neighborhood of a singular point. You can find a proof in Milnor's book "Singular Points of Complex Hypersurfaces", Annals of Mathematics Studies 61, remark at page 13. Notice that $X$ can be a topological manifold (i.e., a ...


22

If $\mathcal{X}$ is a normal Deligne-Mumford stack then its coarse moduli space $X$ is normal. Since $\mathcal{X}$ its normal its admits an ├ętale atlas $U_i\rightarrow\mathcal{X}$, with $U_i$ normal schemes. Now, the statement follows because $U_i$ is normal and $G$ is a finite group acting on $U_i$ then $U_i/G$ is normal as well. In dimension one this ...


14

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 .


13

The difference between the geometric genus of the singularity and the geometric genus of a smoothing (this one being called the arithmetic genus of the singularity) is often called the delta invariant. If $A$ is the local ring of the singularity, $B$ its normalization, then the delta invariant is the dimension of the complex vector space $B/A$. It is rather ...


13

It's a standard result in commutative algebra that every noetherian integral domain is a UFD if and only if every prime ideal of height 1 is principal. When applied to the local rings of X this gives exactly the equivalence above.


13

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, normal, Cohen-Macaulay and has at most cyclic quotient singularities (hence rational singularities), see [2], p. 122. However, weighted projective spaces ...


13

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 $U=X\setminus Z$ (or dually, proving that homology classes are obtained by pushforward from an open subset) are usually called Purity Theorems in algebraic geometry. In ...


12

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 wrote a complicated Noetherian induction in this answer but I just realized this is really easy. Main point: There is a canonical way to find the thickened ...


12

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 pluricanonical rings, there is a klt pair of log general type $(Y',B')$ with isomorphic pluricanonical ring (more precisely $R(K_X)^{(a)}\cong R(K_{X'}+B')^{(b)}$ for ...


11

The Kontsevich recursion formula is not special to $\mathbf{CP}^2$. It is a particular application of the more general associativity formula for quantum cohomology, which is something true for symplectic manifolds in general. Going from associativity to actual numbers counting curves is quite combinatorially involved, I guess, and gets worse as the classical ...


11

I struggled with the Whitney conditions myself. More precisely, I wanted to understand the geometric significance of these conditions. Apparently Whitney was seeking a simple way to characterize equisingularity: allong a connected component of stratum the stratification looks the same. Technically the Whitney conditions are local conditions ...


11

In my opinion a good reference is the book by Greuel, Lossen and Shustin Introduction to Singularities and Deformations. See in particular Theorem 2.46, page 147. In fact, the authors prove the following more precise result. Theorem (Morse Lemma). Let $f \in \mathfrak{m}^2 \in \mathbb{C} \{x_1,\ldots ,x_n \}$ be a germ of holomorphic function, having a ...


11

This is true for real analytic functions $f:R^2\to R$. Let such a function be $f(x,y)$ and we consider the level set $f(x,y)=0$, that is an analytic curve. If at some point $f_x$ or $f_y$ is different from $0$, this point is non-singular and belongs to your set $S$. Now we show that a singular point is a common endpoint of an even number of branches of the ...


11

The rigidity of quotient singularities in dimension greater or equal than $3$ was established by Schlessinger in his paper Rigidity of quotient singularities, Invent. Math. 14 (1971). Roughly speaking, he proved that if $(X, \,x)$ is a local scheme with an isolated singularity at $x$ and $\dim X \geq 3$, then deforming $X$ is equivalent to deform the ...


11

As pointed out by Alex in his comment, this is in general not true. For instance, consider the case $N=d=n=m=2$. Then $|L|$ is the linear system of plane curves of degree $2$ passing through $p_1$ and $p_2$ with multiplicity $2$. Of course there is only one such a curve, namely the line through $p_1$ and $p_2$ counted with multiplicity $2$. In particular, $|...


11

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 $\mathbf{C}^{n+1}=\mathbf{R}^{n+1}\times\mathbf{R}^{n+1}$ we see at once that the link is the Stiefel manifold $\mathrm{V}_2(\mathbf{R}^{n+1})$ of pairs of orthonormal vectors in $\mathbf{R}^{n+1}$...


10

Here is an example: Let $V = \lbrace y^2 = t^2 x^2 + x^3\rbrace \subset \mathbb R^3$. Then the singular set of $V$ is the whole $t$-axis. Let $Y$ be the $t$-axis and $X = V - Y$. Now set $X_1 = X \cap \lbrace x > 0 \rbrace$, $X_2 = X \cap \lbrace x < 0 \rbrace \cap \lbrace t>0 \rbrace$, and $X_3 = X \cap \lbrace x < 0 \rbrace \cap \lbrace t&...


10

The key point is that if $X$ is a locally finite type $\mathbf{C}$-scheme and $x \in X(\mathbf{C})$ then $O_{X,x}$ and $O_{X^{\rm{an}},x}$ are both local noetherian rings with the same completion (induced by the evident canonical map from the algebraic local ring to the analytic one). So for any property of local noetherian rings which is equivalent to ...


10

The weighted projective plane $\mathbb{P}(1,1,n)$ can be viewed as $\mathbb{P}(1,1,n)=\mathbb{C}^3\setminus \{0\}/(x,y,z)\sim (\lambda x,\lambda y,\lambda^n z)$. For $n=1$ we obtain the standard projective plane. For $n>1$, the point $(0,0,1)$ is the unique singular point, and the blow-up of this point is a Hirzebruch surface $\mathbb{F}_n$, with ...


10

Neither Gorenstein nor canonical are preserved. Already Gorenstein is destroyed for the action of $\mathbb{Z}/3\mathbb{Z}$ on $\mathbb{A}^2$ acting by $(x,y)\mapsto (\zeta\cdot x,\zeta\cdot y)$, where $\zeta$ is a primitive cube root of $1$. Also, the whole point of the Reid -- Shepherd-Barron -- Tai criterion is to determine when a quotient singularity is ...


10

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 $\ 0\rightarrow T_X\rightarrow T_{\mathbb{P}^n|X}\rightarrow \mathcal{O}_X(d)\rightarrow \mathcal{E}xt^1(\Omega ^1_X,\mathcal{O}_X)\rightarrow 0$. From this you find easily $H^i(X,...


10

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, $n>1$, the function $(x^2+y^2)^n + x^{2n+1} + y^{2n+1}$ cannot be written in the form $(z^2+w^2)^n$ for any smooth functions $z$ and $w$ of $x$ and $y$.


10

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 interesting in more general cases). This is clear for $\operatorname{Coh}(\operatorname{Crit}(W))$. For matrix factorizations, we can identify $\operatorname{MF}(W)$ ...


9

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 Giant Rat is a function germ $\mathrm{f} : (\mathbb{R}^2,0) \to (\mathbb{R},0)$. We're interested in the set of $(x,y) \in \mathbb{R}^2$, very close to $(0,0) \in \...


9

I don't think so. There are examples of isolated normal threefold singularities that are not Cohen-Macaulay. A hyperplane section is not Cohen-Macaulay, hence it can not be normal, because a normal surface is Cohen-Macaulay.


9

I'm going to assume your singularity is dimension $\geq 3$. Angelo beat me to the answer but he is right, this is not true. But it is true sometimes (including the Cohen-Macaulay case as he implied). A singularity is normal if it is $R1$ and $S2$. In your case, an isolated singularity is normal if the depth at the singular point is at least 2. Now, a ...


9

This algorithm is called the Newton Polygon, and it was really invented and carefuly described by Newton, with examples, see for example, MR1836037 Fischer, Gerd Plane algebraic curves. Translated from the 1994 German original by Leslie Kay. Student Mathematical Library, 15. American Mathematical Society, Providence, RI, 2001. or any other book with a ...


9

If you think about $P(1,2,3)$ as about stack then there is an analogue of the Euler sequence $$ 0 \to O \to O(1) \oplus O(2) \oplus O(3) \to T \to 0. $$ It allows to compute $h^1 = h^2 = 0$ and $h^0 = 5$.


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