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affine or projective algebraic geometry algebraic points algebraically closed field arbitrary assume basis char closure contains Corollary corresponding define definition degree differential form dimension divisor of poles e k[X entire function equation equivalent example exists an element field of constants finite number fixed places formal power series function field genus hypersurface implies inequivalent integer integrally closed intersection multiplicity irreducible affine irreducible k-variety irreducible polynomial irreducible projective curve irreducible projective k-variety isomorphism k-isomorphism Lemma linear series linearly independent mapping maximal ideal non-trivial place non-units non-zero element parallelotope plane curve positive divisor prime ideal projective variety proof of Theorem Proposition 9 prove ramification index rank real numbers repartition representative cone Riemann-Roch Theorem separable extension shows singular point subring subspace uniformizing parameter unique universal domain valuation ring value group vanishes on R(D vector space vp(x Weil-differential zero