Plane Algebraic Curves

abelian algebraic curves analytic set analytic set germ Bézout's theorem braid branches circle coefficients cohomology complex projective compute consider construction corresponding cubic curve of order cusps cycles decomposition defined definition described differential forms divisor double point dual equivalent example fibration finitely formula genus geometry hence homeomorphic homogeneous coordinates homogeneous polynomials homology class infinitely near points inflection points intersection multiplicity intersection numbers intersection points investigation irreducible curve isomorphism Lemma mapping meromorphic Milnor fibre monodromy multiplicity sequence neighbourhood Newton polygon nonsingular normal form obtain P₁ P₂ parabola plane curve polar pole power series preimage projective plane Proof Proposition proved Puiseux expansion Puiseux pairs quadric resolution graph resolution of singularities resp Riemann ring singular points solution space surface tangent tion topological torus knot vanish Weierstrass zero set αβ