Solutions to Hartshorne's Algebraic Geometry/Riemann-Roch Theorem

Exercise IV.1.1 Let have genus . Since is dimension 1, there exists a point , . Pick an . Then for the divisor of degree , (Example 1.3.4), so Riemann-Roch gives . Thus there is an effective divisor such that . Since is degree 0 (II 6.10), has degree , so cannot have a zero of order large enough to kill the pole of of order . Therefore, is regular everywhere except at .