Timeline for Does the invariant from resolution of singularities provide a Whitney stratification?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 31, 2015 at 19:05 | comment | added | Saurabh T | Okay, the reason why I asked you to verify your claim with this example is because this is perhaps the most common example used to explain the conditions a and b. Anyway, I am not sure if your method with 'invariants' will always give a b-stratification. But, if it is of any help to you, any stratification can be 'refined' to a b-stratification. | |
| Jan 31, 2015 at 17:17 | comment | added | Sergio Da Silva | Well in general I am asking whether we get a Whitney b-stratification because I don't know. I can certainly compute the invariant to find that the strata should be the origin, the punctured $z$-axis, and the smooth locus. This should work as a $b$-stratification, but I am less confident about claiming this as fact. | |
| Jan 31, 2015 at 0:04 | comment | added | Saurabh T | Okay, so in the case of Whitney umbrella, the method you described gave a Whitney b-regular stratification. What about the following variety $y^2=z^2x^2+x^3$. Do you still get a Whitney b-stratification? | |
| Jan 30, 2015 at 16:13 | comment | added | Sergio Da Silva | Yes I am referring to the $X_a$'s as the strata for the $b$-stratification (by which I assume you mean a strata that satisfies Whitney condition $b$). For example, you can see fairly easily that if you had a sequence in the smooth locus and a sequence in the $y$-axis converging to a point away from the origin, then secant lines converging to $L$ would certainly be contained in the limit of the tangent planes of the smooth points. You should also convince yourself that this does not happen at the origin if I instead chose my strata to be the entire $y$-axis and the smooth locus. | |
| Jan 30, 2015 at 14:48 | comment | added | Saurabh T | You just described a Whitney b-stratification of the Whitney umbrella. I know that. I wanted to know if it is possible to picture $X_a$'s in this case, ($X_a$'s as in your definition). Or, do you mean $X_a$'s are indeed the strata of this b-stratification of the Whitney umbrella. | |
| Jan 30, 2015 at 14:38 | comment | added | Sergio Da Silva | Well you can draw the Whitney Umbrella using the coordinates I gave, and then you can colour the origin (the pinch point) one colour, the $y$-axis minus the origin (the singular normal crossing points) another colour, and then you can colour the remaining points (all the smooth ones) with a final colour. This would be your Whitney stratification. Of course when working over $\mathbb{C}^3$ you have to draw your picture in $\mathbb{R}^3$ like this en.wikipedia.org/wiki/Whitney_umbrella | |
| Jan 30, 2015 at 13:24 | comment | added | Saurabh T | Is it possible to draw a picture for the collection $X_a$ in the case of the Whitney umbrella? | |
| Jan 22, 2015 at 19:38 | history | asked | Sergio Da Silva | CC BY-SA 3.0 |