Plane Algebraic Curves (Google eBook)
American Mathematical Soc., 2001 - 231 Seiten
The study of the zeroes of polynomials, which for one variable is essentially algebraic, becomes a geometric theory for several variables. In this book, Fischer looks at the classic entry point to the subject: plane algebraic curves. Here one quickly sees the mix of algebra and geometry, as well as analysis and topology, that is typical of complex algebraic geometry, but without the need for advanced techniques from commutative algebra or the abstract machinery of sheaves and schemes.
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Affine Algebraic Curves and Their Equations
The Projective Closure
Tangents and Singularities
Polars and Hessian Curves
The Dual Curve and the Plücker Formulas
The Ring of Convergent Power Series
Parametrizing the Branches of a Curve by Puiseux Series
Tangents and Intersection Multiplicities of Germs of Curves
The Riemann Surface of an Algebraic Curve
affine algebraic curve C C analytic assume Bezout's theorem biholomorphic bitangents C C C2 C(Xi called carrier Chapter coefficients compute connected component construction coordinates Corollary corresponding covering map curve of degree cuspidal cubic cuspidal tangent decomposition define definition degC divisor dual curve equation equivalent examples exist Figure finite genus formula geometric germs of curves given gives Hence Hessian curve homeomorphism homogeneous of degree implicit function theorem inflection points intersection multiplicity invariant Let f linear factors minimal polynomial mult(C neighborhood Newton polygon nodal cubic onlg ordp(C Pi(C plane Pliicker formulas Plucker curve point at infinity points of intersection polar polynomial of degree projective Proof properties prove Puiseux parametrization Puiseux series quadric quartic rational parametrization resultant Riemann surface Section 8.1 simple cusp simple double point singular point smooth points Study's lemma topology unique factorization domain Weierstrass polynomial zeros