Algebraic curves: an introduction to algebraic geometry

affine variety algebraic set algebraic subset algebraically closed assume Bezout's Theorem birationally equivalent called canonical divisor change of coordinates Chapter Char(k closed subvariety comaximal containing COROLLARY curve F curve of degree defined deg(D deg(F denote div(G div(z e k[X,Y element F and G F e k[X finite number flex follows form of degree function field Hint hyperplane hypersurface induced integer intersection number irreducible curve isomorphism LEMMA Let F linear maximal ideal module-finite morphism Noetherian non-singular non-singular cubic non-zero Nullstellensatz Op(V open subvariety ordinary multiple points point on F polynomial F polynomial map prime ideal Problem projective change projective plane curve projective variety Proof Prop PROPOSITION quadratic transformation quotient field R-module rational function rational map residue resp ring homomorphism Show that F simple point subring Suppose tangent line tangent to F topology uniformizing parameter unique vector space zero