Plane Algebraic Curves

1-dimensional abelian algebraic curves analytic set analytic set germ Bézout's theorem braid choose circle coefficients compute consider corresponding cubic curve germs curve of order cycles decomposition defined definition described differential forms dimension divisor double point dual equivalent example fibration fibre finitely follows formula genus geometry hence holomorphic functions homeomorphic homogeneous coordinates homogeneous polynomials homology class infinitely near points inflection points intersection multiplicity intersection number intersection points investigation irreducible components isomorphism Lemma manifold mapping Milnor neighbourhood Newton polygon nonsingular normal form o-processes obtain P₁ P₂ parabola plane curve poles polydisc preimage projective plane Proof Proposition prove Puiseux expansion Puiseux pairs quadric resolution resolution of singularities resp Riemann singular points solution space surface tangent tion topological torus knot triangle TURK'S-HEAD vanish Weierstrass zero set αβ ду