Timeline for Whitney Conditions vs Equisingularity
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9 events
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| Mar 9, 2017 at 14:45 | comment | added | Liviu Nicolaescu | I have to admit that I am not very fluent in algebraic geometry, but geometrically the notions of equisingularity should be, in spirit, very similar. I am aware that "in spirit" is not the answer you're looking for. I don't think Whitney's condition has an exact counterpart in algebraic geometry since it involves metric notions which are far removed from Zariski type topologies. I would look for definitions that have the attributes "local" and "flat" in them. | |
| Mar 9, 2017 at 14:41 | comment | added | Aswin | (..condt...) My question was really about understanding how far away Zariski's notion of equisingularity is from Whitney's notion. I am not questioning the fact that to retrieve Whitney's notion of equisingularity.. one needs both conditions A and B. I find Whitney's conditions very intuitive and I would be quite happy if I never have to think of another notion equisingularity .. but I ran into the term "equisingular" in some situations where there was no Whitney stratification. So, I wanted to be sure how exactly Zariski's theory diverged from Whitney's theory .. | |
| Mar 9, 2017 at 14:37 | comment | added | Aswin | Those are some very nice notes and a fantastic introduction to the Whitney theory! But, unless I have misunderstood your answer and comments.. these are in a slightly different direction from my question. I believe Zariski's definition of equisingularity is in general different from the one Whitney obtained using his two conditions (the other answer by Libli + ref to Tessier's works seems to also indicate this). | |
| Mar 9, 2017 at 14:21 | comment | added | Liviu Nicolaescu | Heres a bit of history. Initially Whitney believed that A guarantees equisingularity. He then observed that the Whitney cusp with an obvious stratification satisfies A yet it is not equisingular. That is when he improved A to B. I suggest you have of look at Section 4.2 of the notes www3.nd.edu/~lnicolae/Morse2nd.pdf It is a rather informal introduction to Whitney stratifications with pictures, examples and a discussion of equisingularity that might address some of your questions. It also contains references I found useful. | |
| Mar 9, 2017 at 13:00 | comment | added | Aswin | Yes, I am aware that B => A for Whitney conditions. I was asking a question about equisingular stratifications that are not Whitney. Do these stratifications obey condition A or do they not even obey that ? | |
| Mar 9, 2017 at 12:15 | comment | added | Liviu Nicolaescu | It is known that B implies A. Condition A does not guarantee equisingularity The Whintey cusp $y^2+x^3-x^2z^2=0$ satisfies $A$ along the $z$ axis, but not $B$. | |
| Mar 9, 2017 at 11:54 | comment | added | Aswin | In this case, I think the first Whitney condition (usually called condition $A$) is still obeyed.. right ? In general, does the condition of equisingularity require that atleast the first Whitney be obeyed or is even this not a necessary requirement ? | |
| Mar 4, 2017 at 22:12 | history | edited | Liviu Nicolaescu | CC BY-SA 3.0 |
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| Mar 4, 2017 at 22:05 | history | answered | Liviu Nicolaescu | CC BY-SA 3.0 |