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Can anybody explain why the vector bundle corresponding to a locally free sheaf F is global spec of sym of the dual of F and not just F? How does a section get identified with a polynomial in the dual?

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I suggest you work it out when the base scheme is Spec of a field. – Laurent Moret-Bailly Nov 17 '10 at 20:32
Linear functions on V are elements of V^*, so polynomial functions on V are elements of Sym(V^*). – Dustin Clausen Nov 17 '10 at 21:21
Functoriality . – Martin Brandenburg Nov 17 '10 at 23:08
Or maybe when you say "corresponding to" you're thinking of a different correspondence from the rest of us. – Tom Goodwillie Nov 18 '10 at 11:38

Let $L$ be a locally free ${\cal O}_X$-module of finite rank. Define $V=Spec(Sym(L^\vee))$. Then $$Mor_X(X, V)={\cal O}_X-Alg(Sym(L^\vee), {\cal O}_X)=Hom(L^\vee, {\cal O}_X)=L(X).$$ The universal mapping property of the (global) Spec is in EGA II.1.

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