Tuesday, 27 June 2006

ag.algebraic geometry - Minimal size of an open affine cover

This is not a complete answer by any means, but here are the two most basic arguments. First of all, you have that every projective scheme that can be embedded in mathbbPn can be covered by n+1 open affines, namely the closed subschemes of the affines Ui=lbracezineq0rbracecongmathbbAn.



For a lower bound, think Cech-cohomologically: if X can be covered by k affine opens, then checkHl(X)=0 for every l>k. If X is Noetherian separated, then Cech cohomology coincides with sheaf cohomology, which indicated that you need at least maxlbracel;|;Hl(X)neq0rbrace open affines to cover it.

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