MIT OpenCourseWare http ocw mit edu 18 727 Topics in Algebraic Geometry Algebraic Surfaces Spring 2008 For information about citing these materials or our Terms of Use visit http ocw mit edu terms Homework 1 18 727 Spring 2008 1 Do the blowups necessary to reslove the the du Val singularities a A4 x2 y 2 z 5 0 b D5 x2 y 2 z z 4 0 c E6 x2 y 3 z 4 0 d E7 x2 y 3 yz 3 0 e E8 x2 y 3 z 5 0 2 Show that every locally free sheaf of rank n on P1 is a isomorphic to a direct sum of n line bundles Hint choose an invertible subsheaf of maximal degree 3 Prove the following proposition Let X B be a morphism from a nonsingular projective surface to a nonsingular projective curve and let D ni Ei be a ber of Then for every divisor D n i Ei with n i Z i e supported on the ber we have D 2 0 If the ber D is connected then D 2 0 if and only if a Q such that D aD 1
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