Abstract
Let A be the coordinate ring of an affine piece of a smooth curve, V , defined over either R or an algebraically closed field k. We ask which maximal ideals of A are principal. We give a complete determination if V has genus 0 or 1, and give partial results if V has genus >/ 2. We conjecture that if k is algebraically closed of characteristic 0, genus ( V) >/ 2, then A has only finitely many principal maximal ideals. This conjecture is equivalent to the Mordell Conjecture of Diophantine Geometry.
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