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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
(2.3.7.1)
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:
(2.3.7.2)
R"f, (ex)
~ , R "+1 n, (Or( d)).
Using the standard
covering of projective space, the ~smodule
R"+I n, (Or(d)) is easily computed:it's the free Csmodule
(2.3.7.3)
"forms" of degree  d in
/the Cgsspan 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.
 Fall '11
 NormanKatz

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