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 + y2 + z5 = 0, (b) D5 : x2 + y2z + z4 = 0, (c) E6 : x 2 + y 3 + z 4 = 0, (d) E7 : x2 + y3 + yz3 = 0, (e) E8 : x2 + y3 + z5 = 0. 2. Show that every locally free sheaf of rank n on P1 is a isomorphic to a dir e c t sum of n line bundles. (Hint: choose an invertible subsheaf of maximal degree.) 3. Prove the following proposition: Let π : X → B be a morphism from a (nonsingular projective) surface to a (nonsingular projective) curve, and let D = � niEi be a fiber of π. Then for every divisor D ′ = � n ′ iEi with n ′ i ∈ Z (i.e. supported on the fiber), we have (D ′2) ≤ 0. If the fiber D is connected then (D ′2) = 0 if and only if ∃a ∈ Q such that D ′ = aD.
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