Timeline for Semi-continuity of intersection numbers
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| May 6, 2016 at 12:57 | history | edited | Giulio | CC BY-SA 3.0 |
I made the question more precise
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| May 6, 2016 at 12:51 | comment | added | Giulio | I just found another relevant post! I'll updated my question | |
| May 5, 2016 at 21:14 | comment | added | Giulio | I am happy with the fact that the intersection number can go up, a bit less happy with the fact that it can go down. (I have in mind the example of the dimension of the fibre of a morphism: the dimension can only go up under specialisation, not down). For instance, take the divisor to be effective. I do not quite understand the example with divisor with negative self-intersection. | |
| May 5, 2016 at 21:06 | comment | added | Jason Starr | @FanZheng. Yes, that is even simpler. | |
| May 5, 2016 at 20:10 | comment | added | Fan Zheng | @JasonStarr Is it even simpler if we just take a single (closed) point in $P^1\times\Delta$? | |
| May 5, 2016 at 19:57 | comment | added | Jason Starr | ... or just use my example, but with -1 coefficients on the divisors. | |
| May 5, 2016 at 19:56 | comment | added | Jason Starr | Anyway, if you want a contradiction to upper semi-continuity, consider the same type of example with $D_1(t) = D_2(t) = D(t)$ a family of divisors on a surface with negative self-intersection. | |
| May 5, 2016 at 19:54 | comment | added | Jason Starr | It contradicts lower semi-continuity. | |
| May 5, 2016 at 19:47 | history | edited | Giulio | CC BY-SA 3.0 |
added 228 characters in body
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| May 5, 2016 at 19:45 | comment | added | Giulio | this example does not contradict a semi-continuity statement about intersection numbers. Anyway, I will make more precise my question | |
| May 5, 2016 at 17:03 | comment | added | Jason Starr | In $\mathbb{P}^1$ with homogeneous coordinates $[x,y]$, consider the closed subscheme in $\mathbb{P}^1\times \Delta$ with defining ideal $\langle x^2,tx\rangle$. That is a closed subscheme that is not flat over $\Delta$. However, its intersection with the fiber over every $a\in \Delta$ is a divisor in $\mathbb{P}^1$. Except if $a=0$, that divisor has degree $1$. For $a=0$, the divisor has degree $2$. Most algebraic geometers would not consider the closed subscheme to be a "family of divisors", i.e., they would impose a flatness hypothesis (or some hypothesis equivalent to flatness). | |
| May 5, 2016 at 15:26 | history | asked | Giulio | CC BY-SA 3.0 |