Question on a projective bundle associated to a vector bundle - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T04:55:23Z http://mathoverflow.net/feeds/question/72671 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/72671/question-on-a-projective-bundle-associated-to-a-vector-bundle Question on a projective bundle associated to a vector bundle Mohammad F.Tehrani 2011-08-11T10:29:41Z 2011-08-11T11:07:39Z <p>Following is quoted from -Nakayama, On Weierstrass models-, </p> <p>" Let $S$ be a complex surface, and $L$ a line bundle on it. consider $P=\mathbb{P}(\mathcal{O}_S\oplus L^2 \oplus L^3)$. Let $a$ and $b$ be arbitrary sections of $L^{-4}$ and $L^{-6}$ and let $(x,y,z)$ be the canonical sections of $\mathcal{O}_P(1)\otimes L^{-2}$, $\mathcal{O}_P(1)\otimes L^{-3}$ and $\mathcal{O}_P(1)$ respectively which correspond to the natural injections of $L^2$, $L^3$ and $\mathcal{O}_S$ into $\mathcal{O}_S\oplus L^2 \oplus L^3$. Then the Weierstrass model is given by equation $y^2z=x^3+axz^2+bz^3$ in $P$ and is an elliptic fibration over $S$ ..."</p> <p>My question: When I do calculations for my self, I see that the embeddings given above should correspond to canonical sections of $\mathcal{O}_P(1)\otimes L^{2}$, $\mathcal{O}_P(1)\otimes L^{3}$ and $\mathcal{O}_P(1)$, not the one he is saying and therefore we should consider $(a,b)$ as sections in dual of what he has said and at the end we should get an equation in $\mathcal{O}(3)\otimes L^6$ not in $\mathcal{O}(3)\otimes L^{-6}$ ? How does he get $x$ (and similarly $y$) from embedding $L^2 \rightarrow \mathcal{O}_S \oplus L^2 \oplus L^3$??</p> <p>My calculation: We have the exact sequnce $$0\rightarrow \mathcal{O}_P(-1) \rightarrow \mathcal{O}\oplus L^2 \oplus L^3 \rightarrow Q \rightarrow 0$$ </p> <p>over $P$, from which we get </p> <p>$$0 \rightarrow Hom(\mathcal{O}(-1),\mathcal{O}(-1)) \rightarrow Hom(\mathcal{O}(-1),\mathcal{O}\oplus L^{2} \oplus L^{3}) \rightarrow T_{W/S} \rightarrow 0$$</p> <p>Here the last object is the relative tangent bundle and the first map is given by three sections $(z,x,y) \in Hom(\mathcal{O}(-1),\mathcal{O}\oplus L^{2} \oplus L^{3})\cong \Gamma(\mathcal{O}(1)) \oplus \Gamma(\mathcal{O}(1)\otimes L^{2}) \oplus \Gamma(\mathcal{O}(1)\otimes L^{3})$ as above. These are analogue of $(x,y,z)$ coordinate on projective space.</p> http://mathoverflow.net/questions/72671/question-on-a-projective-bundle-associated-to-a-vector-bundle/72673#72673 Answer by a-fortiori for Question on a projective bundle associated to a vector bundle a-fortiori 2011-08-11T10:55:51Z 2011-08-11T11:07:39Z <p>There are two competing definitions for <code>$\mathbb P(\mathcal E)$</code>, one classifies subbundles of <code>$\mathcal E$</code> of rank 1, the other classifies quotients. With the latter (as used in EGA or Hartshorne), you have a canonical quotient <code>$p^*\mathcal E\to\mathcal O(1)$</code> instead of a canonical subbundle <code>$\mathcal O(-1)\to p^*\mathcal E$</code>. As long as <code>$\mathcal E$</code> is locally free of finite rank, there is no big difference, you can just dualize <code>$\mathcal E$</code> to pass from one to the other.</p>