Timeline for Calculating chern numbers yields a contradiction, why?
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9 events
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| Mar 16, 2012 at 23:26 | vote | accept | Jesko Hüttenhain | ||
| Mar 16, 2012 at 23:26 | comment | added | Jesko Hüttenhain | Well. No. Now, all I've said seems stupid to me, but I suppose that's what they call learning. Thanks for putting up with it. | |
| Mar 16, 2012 at 22:00 | comment | added | Johannes Nordström | But $[H]=[H']$ does not imply $[P] = [P']$. | |
| Mar 16, 2012 at 17:52 | comment | added | Jesko Hüttenhain | More generally, I feel that $[P]=f^\ast[H']=f^\ast[H]=[P]+[E]$ is a contradiction. | |
| Mar 16, 2012 at 12:16 | comment | added | Jesko Hüttenhain | It is beginning to become more clear to me, slowly. I have one more question, though: if $H$ is a hyperplane containing $X$ and $H'$ is a generic one, intersecting $X$ transversally, then we should have $[H]=[H']$ as classes in the Chow ring. Denoting by $P$ and $P'$ the proper transforms, this would imply that $[P']^2=0$ as well. That would mean that all proper transforms of hyperplanes suddenly have self-intersection zero. That sounds weird to me. | |
| Mar 14, 2012 at 8:48 | comment | added | Johannes Nordström |
$f^*(c_1(Y)^2c_2(Y))$ has the same degree as $c_1(Y)^2c_2(Y)$, but if $f^*c_i(Y)$ equaled $c_i(\tilde Y)$ the calculation would be trivial. Certainly $c_1(Y)^4$ is not equal to $c_1(\tilde Y)^4$. The formula you use for the intersections of $P$ and $E$ requires that $P$ be the proper transform not of a generic hyperplane, but of one that contains $X$. Two such hyperplanes intersect along $X$, so when you blow up $X$ their proper transforms are disjoint.
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| Mar 13, 2012 at 23:59 | comment | added | Jesko Hüttenhain | Also, what exactly is your reason for $P^2=0$? | |
| Mar 13, 2012 at 23:52 | comment | added | Jesko Hüttenhain | The fact that the degree of a zero-cycle is preserved should follow (in my opinion) from example 1.7.4 in Fulton. Note that a finite surjective morphism between nonsingular varieties is always flat. By Corollary 6.7.2 in Fulton, since a generic hyperplane intersects $X$ in dimension $n-1-d$, I concluded that the pull-back of a hyperplane clas is its strict transform. Where is my error? | |
| Mar 13, 2012 at 17:35 | history | answered | Johannes Nordström | CC BY-SA 3.0 |