It is well known that the Grassmanian of lines in $\mathbb P^3$ is isomorphic to a quadric in $\mathbb P^5$. I would like to ask if the tautological rank two bundle on the grassmanian extends to a rank two bundle on the whole of $\mathbb P^5$?
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2 Answers
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I would say it doesn't. The tautological Chern classes generate $H^*$ of the Grassmanian, but the inclusion to $\mathbb{P}^4$ does not induce an epimorphism.
answered Feb 21, 2014 at 11:14
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$\begingroup$ Yes, sure. Actually, any $\mathbb{P}$ :) $\endgroup$ Commented Feb 21, 2014 at 11:29
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$\begingroup$ Alex, I think your reasoning is correct. Could you give me a reference to some nice book that explains that tautological Chern classes generate $H^*$ of the Grassmanian? $\endgroup$ Commented Feb 21, 2014 at 11:31
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$\begingroup$ Over $\mathbb{Q}$ this is just comparing the dimensions (the keyword being "Schubert calculus"). A lot about Grassmanians (including everything about your favorite one) is found here: David Eisenbud and Joe Harris, 3264 & All That: Intersection Theory in Algebraic Geometry $\endgroup$ Commented Feb 21, 2014 at 11:43
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Here is an answer for this specific case. Let $Q$ be the tautological bundle on $X$, the Grassmannian. If it can be extended to say $E$ on 5-spcae, using the fact that $H^1_*(Q)=0$, one gets from the exact sequence, $0\to E(-2)\to E\to Q\to 0$, that $H^1_*(E)=0$ and thus $H^0(E)\to H^0(Q)$ is onto. In particular, you can lift the 4 sections of $Q$ to $E$ and since they generate $Q$, we see that they generate $E$ outside finite set of points. Restricting to a general hyperplane not passing through these points, we get a 4 generated rank 2 bundle on 4-space, which by a theorem of Faltings (see Inventiones, 1981), is trivial. Then the usual Horrocks theorem will say that $E$ itself is trivial and thus so is $Q$, which is absurd.
