## Geometry of Algebraic Curves, Volume 1In recent years there has been enormous activity in the theory of algebraic curves. Many long-standing problems have been solved using the general techniques developed in algebraic geometry during the 1950's and 1960's. Additionally, unexpected and deep connections between algebraic curves and differential equations have been uncovered, and these in turn shed light on other classical problems in curve theory. It seems fair to say that the theory of algebraic curves looks completely different now from how it appeared 15 years ago; in particular, our current state of knowledge repre sents a significant advance beyond the legacy left by the classical geometers such as Noether, Castelnuovo, Enriques, and Severi. These books give a presentation of one of the central areas of this recent activity; namely, the study of linear series on both a fixed curve (Volume I) and on a variable curve (Volume II). Our goal is to give a comprehensive and self-contained account of the extrinsic geometry of algebraic curves, which in our opinion constitutes the main geometric core of the recent advances in curve theory. Along the way we shall, of course, discuss appli cations of the theory of linear series to a number of classical topics (e.g., the geometry of the Riemann theta divisor) as well as to some of the current research (e.g., the Kodaira dimension of the moduli space of curves). |

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### Contents

3 | |

4 Abelian Varieties and the Theta Function | 20 |

6 A Few Words About Moduli | 28 |

APPENDIX | 50 |

1 Applications of the Discussion About Plane Curves with Nodes | 56 |

Geometric Description | 67 |

4 Determinantal Varieties and Porteous Formula | 83 |

5 A Few Applications and Examples | 93 |

5 First Consequences of the Infinitesimal Study of G C and WC | 191 |

CHAPTER V | 203 |

CHAPTER VI | 225 |

2 Kempfs Generalization of the Riemann Singularity Theorem | 239 |

3 The Torelli Theorem | 245 |

APPENDIX | 281 |

3 Theta Characteristics | 287 |

APPENDIX C | 295 |

CHAPTER III | 107 |

2 Castelnuovos Bound Noethers Theorem and Extremal Curves | 113 |

3 The EnriquesBabbage Theorem and Petris Analysis of the Canonical Ideal | 123 |

CHAPTER IV | 153 |

2 The Universal Divisor and the Poincaré Line Bundles | 164 |

3 The Varieties W C and G C Parametrizing Special Linear Series on | 176 |

4 The Zariski Tangent Spaces to G C and WC | 185 |

CHAPTER VII | 304 |

3 The Connectedness Theorem | 311 |

CHAPTER VIII | 330 |

2 Three Applications of the GrothendieckRiemannRoch Formula | 333 |

5 Diagonals in the Symmetric Product | 358 |

374 | |

### Common terms and phrases

assume base point base-point-free bi-elliptic birational Brill-Noether canonical curve Castelnuovo's Chapter Chern class codimension Cohen–Macaulay cohomology component compute conclude containing coordinates curve of degree curve of genus defined denote determinantal variety dimension divisor of degree effective divisor elliptic equation equivalent extremal curves fiber fundamental class genus g geometric given hence Hint holomorphic hyperelliptic hyperplane ideal integer irreducible isomorphism Jacobian lemma linear series linear system locus matrix meromorphic function moduli monomials morphism nodes notations pencil plane curve Poincaré line bundle polynomial Porteous formula preceding exercise projectively normal proof Proposition prove pull-back quadrics rank rational normal curve rational normal scroll Riemann surface Riemann–Roch theorem sheaf Show singularity theorem smooth curve smooth plane quintic special divisors subspace subvariety Suppose symmetric tangent cone tangent space theta characteristics theta divisor trigonal vanishes vector bundle vector space Weierstrass Zariski tangent space zero