Dr. Katz DEq Homework Solutions 34

Dr. Katz DEq Homework Solutions 34 - 34 N.M. Katz: a t...

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34 N.M. Katz: at least locally on S. Localizing on S, we may and will assume that X is defined by a homogeneous form of degree d, H = H(X1 . .... X,+ 2)e F(S, C s) [X1 . .... X,+2]. The corresponding short exact sequence on P d~,,p~ + 1 ( 0---~ (_Or(-d) ~, d~ e --* C x ~ 0 gives an isomorphism of cohomology sheaves on S (n: P; § 1 ~ S denoting the projection) via coboundary: ( R"f, (ex) ~ , R "+1 n, (Or(- d)). Using the standard covering of projective space, the ~s-module R"+I n, (Or(-d)) is easily computed:it's the free Cs-module ( "forms" of degree - d in /the Cgs-span of those monomials --1 --1 W Wl Wn+2 " (~s[X1 . .... Xn+2 X 1 . .... Xn+ 2]/ X = X 1 .
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This note was uploaded on 12/21/2011 for the course MAP 4341 taught by Professor Normankatz during the Fall '11 term at UNF.

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