Unformatted text preview: ( x ) 6 = ∞e x ( f )1 if f ( x ) = ∞ . (8.20) Here df is the meromorphic diﬀerential deﬁned locally by df dt dt , where t is a local parameter at x . Since the degree of df is ﬁnite we obtain that there are only ﬁnitely many points x ∈ X such that e x ( f ) > 1. In particular, there is a ﬁnite subset of points S = { y 1 ,...,y s } in P 1 ( C ) such that, for any y 6∈ S X x : f ( x )= y e x ( f ) = n = # f1 ( y ) . (8.21) Taking into account the formulas ( ?? )(8.21), we obtain 2 g2 = X x ∈ X ν x (div( df )) = X y : f ( y ) 6 = ∞ ( e x ( f )1) + X y : f ( y )= ∞ (e x ( f )1) = X x ∈ X ( e x ( f )1)2 X f ( x )= ∞ e x = X x ∈ X ( e x ( f )1)2 n = X y ∈ Y ( n# f1 ( y ))2 n. (8.22)...
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This note was uploaded on 01/08/2012 for the course MATH 300 taught by Professor Ontonkong during the Fall '09 term at SUNY Stony Brook.
 Fall '09
 ONTONKONG

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