Timeline for When is a scheme a zero-set of a section of a vector bundle?
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| Oct 1, 2010 at 7:00 | comment | added | Timo Schürg | Sorry, the rank of the vector bundle in the comment should be $rk(E^{-1})$. | |
| Oct 1, 2010 at 6:59 | comment | added | Timo Schürg | Thanks for the answer! The way I posed the question was really for arbitrary rank of $E$.I also figured out another way of proving your statement in the meantime: Just take the first step of a locally free resolution of the ideal sheaf of $X$ in $M$. That gives a surjective morphism $E \to I$, and thus a section. The case I really cared about was when $X$ has a perfect obstruction theory $E^{-1} \to E^{0}$. I wanted to fix $dim(M)=rk(E^{0})$, and the rank of the vector bundle to be $rk(E^{0})$. If $X$ is affine, that actually works! It's Appendix A of front.math.ucdavis.edu/1001.2719. | |
| Sep 30, 2010 at 12:47 | history | answered | Angelo | CC BY-SA 2.5 |