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Results tagged with singularity-theory
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user 2083
Singularities in algebraic/complex/differential geometry and analysis of ODEs/PDEs. Singular spaces, vector fields, etc.
5
votes
Accepted
Computations of divisor class monoids
Here are a few remarks about $DC(A)$ (assuming $A$ is a complete Noetherian local domain of dimension $1$).
The equivalence relation in $D(A)$ is just isomorphism as $A$-modules. So you can view $DC( …
- 30.6k
0
votes
on the relative conductor of curve singularity and quotient of ideals
Ok, so it has been 7 years, but I do have something new to add to the answers by Karl and Sándor.
All your questions are about whether some modules/ideals are reflexive. That is because for a fraction …
- 30.6k
4
votes
Accepted
Relationship between Hilbert-Samuel multiplicity and polar multiplicity
First, the quantity you defined is the Hilbert-Samuel multiplicity of the ideal $J= (f_x,f_y)$ in $R=\mathbb C[[x,y]]/(f)$. The multiplicity of $f$ usually refers to the multiplicity of the maximal id …
- 30.6k
5
votes
Accepted
Factoriality vs $\mathbf{Q}$-factoriality for threefolds hypersurfaces with isolated singula...
The answer to Question 2 is true. If $(R,m)$ is a local complete intersection of equicharacterstic $0$ and of dimension $3$ then the Picard group of $Y = \text{Spec} R-\{m\}$ is torsion-free. This gr …
- 30.6k
5
votes
Q-factorial and rational singularities on surfaces
About the converse: one does need all the assumptions Karl mentioned in his answer. There are $2$-dim. complete local rings which are UFD but does not have rational singularity. One such example (due …
- 30.6k
6
votes
What formal properties should resolution of singularities have?
It should exist in all characteristics, even mixed characteristic!
This sounds somewhat cheeky, but I was fairly serious. To algebraists, and the OP is one last time I met him, a purely ring-the …
- 30.6k
1
vote
Tame ramification of (mild) curve singularities.
Here is a reference that may be relevant (you may know it already): Auslander-Rim has a paper called "Ramification index and multiplicity" . Even though they mostly discussed normal rings, their defin …
- 30.6k