Introduction to Algebraic Geometry

abstract variety affine representative affine variety algebraic extension algebraic set algebraically closed algebraically independent assume automorphism basis biholomorphic birational correspondence Chapter closure codimension coefficients components concludes the proof conjugate contained coordinate ring COROLLARY denote differential form dimension theorem discrete valuation element equal equation everywhere defined exists field of definition finite number function field Hence homomorphism hypothesis ideal in k[X induced integrally closed intersection isomorphism k-normal k-variety lemma Let f linear system linearly disjoint linearly independent local ring locus Math maximal ideal monomials non-singular normal P₁ pole polynomial prime divisor prime ideal projective variety purely inseparable quotient field rational map regular extension ring k[x separable algebraic separable extension simple point smallest field specialization subfield subvariety suffices to prove Suppose theory transcendental trivial unique valuation ring vanishes variety defined vector space w₁ write x₁ y₁ zero α α α