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6
votes
2answers
558 views

Is the desingularization of a normal variety with only quotient singularities projective

The base field will be the field of complex numbers. I have a slightly technical problem concerning the resolution of singularities of a certain variety. Basically, I want to to know if it is ...
3
votes
3answers
412 views

The link of a singular quintic hypersurface in CP^4

Given a family of quintic hypersurfaces in $\mathbb{CP}^4$ by $x_1^5+x_2^5+x_3^5+x_4^5+x_5^5+(5+\epsilon)x_1x_2x_3x_4x_5$ we get a singular variety for $\epsilon=0$ with 125 singular points. I know ...
6
votes
3answers
868 views

Is there an obvious way for showing singularities are quotient?

I'm stuck on a technicality concerning singularities. Basically, I have to show that the singularities of a $\mathbf{certain}$ normal projective variety over $\mathbf{C}$ are rational. (I won't ...
4
votes
1answer
482 views

When singular points of a reduced scheme are not dense in it?

A stupid AG question: could singular (Zarisky) points be dense in a reduced (Noetherian) scheme $S$? If yes, which 'standard' restrictions on $S$ could ensure that this does not happen? For example, ...
3
votes
2answers
1k views

On minimal resolution of singularities and the type of singularities

Let $Y$ be a normal projective surface, let $X$ be a smooth projective surface and let $\pi:Y\longrightarrow X$ be a finite morphism. Why are all singularities of $Y$ cyclic quotient singularities? ...
3
votes
4answers
906 views

Matrix factorization categories for ADE singularities

What is known about the matrix factorization categories of singularities of type ADE? Any references on this would be greatly appreciated. Background: For ADE singularities, see for example this. For ...
6
votes
3answers
727 views

What does being Analytically Isomorphic imply for classification of singularities on curves?

Hartshorne I.5 mentions the definition of being analytically isomorphic: P on X and Q on Y are analytically isomorphic iff the completion of O_P is isomorphic to the completion of O_Q where the ...
18
votes
5answers
2k views

Is a 'generic' variety nonsingular? Or singular?

I'd like to know whether there's some coherent meaning of 'generic' for which one can say that a 'generic' variety over an algebraically closed field $K$, say, is nonsingular or singular. We could ...
6
votes
1answer
693 views

Localization of vanishing cycles

Consider a regular holonomic D-module (or a perverse sheaf) $M$ on a smooth variety $X$. Let $f:X\to A^1$ be a polynomial (or holomorphic) function. Question: Is it true that the $\lambda \in A^1$ ...
26
votes
2answers
1k views

Limit of a series of singularities

The $A_\infty$ and $D_\infty$ plane curve singularities have defining equations $x^2=0$ and $x^2y=0$. These equations are "clearly" natural limiting cases of the equations for $A_n$ singularities ...
2
votes
1answer
191 views

Existence of smoothing of Calabi-Yau cones over $dP_{1}$ and $dP_{2}$

The blowdown of the zero section of the canonical bundle of the first del Pezzo surface $dP_{1}$, the blowup of $CP^{2}$ at one point, is a Calabi-Yau cone. I was just wondering if this cone admitted ...
5
votes
1answer
590 views

Resolution of Singularities, Nature of

Hironaka's theorem guarantees an existence of resolution of singularities in characteristic 0. If I am not wrong, it also guarantees (or at least some other result does), that if the resolution is a ...
2
votes
1answer
500 views

Corank 4 hypersurface singularities

A function f: ($\mathbb{C}^n$,0) $\to$ ($\mathbb{C}$,0) is considered a hypersurface singularity if the point $(0,0,\dots,0)$ is the only point in the ideal $\langle \frac{\partial f}{\partial x_1}, ...
3
votes
0answers
449 views

Sebastiani-Thom isomorphism for D-modules

Considering $f:X\to \mathbb{C}$, $g:X\to \mathbb{C}$ and $f\oplus g:(x,y)\mapsto f(x)+g(y)$. The Sebastiani-Thom isomorphism is an isomorphism $\Phi_{f\oplus g}(M\boxtimes N) = \Phi_{f}(M) \otimes ...
4
votes
1answer
426 views

Singularity theory references

I am looking for some good references on singularity theory. I'm interested in singularity theory in the context of mirror symmetry, so this means I'm interested in things like Picard-Lefschetz ...