Timeline for Is there a relation between the first Chern class of a sub canonical submanifold of the complex projective space and the degrees of the polynomials that define locally the submanifold?
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| Oct 2, 2012 at 17:14 | comment | added | Sasha | Of course knowing the generators is not enough, but if you know all the syzygies it may help. For example, in case of $Gr(2,5)$ the resolution has form $O(-5) \to O(-3)^5 \to O(-2)^5 \to O$. From this you see immediately that $\det N^* = O(-5)$ which does the job. Of course we are lucky here that the resolution is so simple. Alternatively, we could argue that there is an exact sequence $0 \to \Lambda^2N^* \to O(-3)^5 \to O(-2)^5 \to N^* \to 0$, which also allows to compute the degree of $N^*$. | |
| Oct 2, 2012 at 13:53 | comment | added | Damian Rössler | I agree but that is because it is easily computable in that case. How do you propose to compute the normal bundle of a general smooth submanifold, if you know some generators of its homogenous ideal ? | |
| Oct 2, 2012 at 13:51 | comment | added | Sasha | @Damian: Even for complete intersections you compute the canonical class by computing the degree of the normal bundle first. | |
| Oct 2, 2012 at 10:20 | comment | added | Damian Rössler | I don't think that the normal bundle is any easier to compute than the canonical one... | |
| Oct 2, 2012 at 10:00 | history | answered | Sasha | CC BY-SA 3.0 |