## Plane Algebraic CurvesPlane Algebraic Curves is a classroom-tested textbook for advanced undergraduate and beginning graduate students in mathematics. The book introduces the contemporary notions of algebraic varieties, morphisms of varieties, and adeles to the classical subject of plane curves over algebraically closed fields. By restricting the rigorous development of these notions to a traditional context the book makes its subject accessible without extensive algebraic prerequisites. Once the reader's intuition for plane curves has evolved, there is a discussion of how these objects can be generalized to higher dimensional settings. These features, as well as a proof of the Riemann-Roch Theorem based on a combination of geometric and algebraic considerations, make the book a good foundation for more specialized study in algebraic geometry, commutative algebra, and algebraic function fields. Plane Algebraic Curves is suitable for readers with a variety of backgrounds and interests. The book begins with a chapter outlining prerequisites, and contains informal discussions giving an overview of its material and relating it to non-algebraic topics which would be familiar to the general reader. There is an explanation of why the algebraic genus of a hyperelliptic curve agrees with its geometric genus as a compact Riemann surface, as well as a thorough description of how the classically important elliptic curves can be described in various normal forms. The book concludes with a bibliography which students can incorporate into their further studies. Book jacket. |

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### Contents

CONTENTS | 1 |

Chapter | 12 |

THE LOCAL RING AT A POINT | 43 |

Chapter 5 | 52 |

Chapter 6 | 70 |

Chapter 7 | 78 |

EXAMPLES OF RATIONAL CURVES | 88 |

Chapter 9 | 97 |

Chapter 10 | 122 |

THE DIVISOR OF A FUNCTION HAS DEGREE O | 128 |

RIEMANNS THEOREM | 133 |

Chapter 15 | 148 |

Chapter 17 | 163 |

Chapter 19 | 179 |

Chapter 21 | 202 |

217 | |

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### Common terms and phrases

affine curve affine plane curve affine subcurve algebraic algebraically closed Assume automorphism called centered Chapter char(k choose compute conclude consider construction containing coordinate ring Corollary correspondence Dedekind domain defined definition degree denote described determines discussion distinct points divides divisor easy element embedding equality equation equivalent Example Exercise exists extension fact factor field flex function field genus given gives hence holds homogeneous homomorphism implies induced infinite integral closure intersection irreducible isomorphism k-algebra Lemma linear maximal ideal means morphism multiplicity noetherian nonsingular projective nonzero prime polynomial prime ideal projective completion projective curve projective plane curve Proof Proposition prove quotient field rational map reader relation Remark Res(f,g resp respect result ring satisfying Show simple point singularity space subset Suppose tangent line Theorem unique valuation vanishes viewed write zero