## Lectures on Curves on an Algebraic SurfaceThese lectures, delivered by Professor Mumford at Harvard in 1963-1964, are devoted to a study of properties of families of algebraic curves, on a non-singular projective algebraic curve defined over an algebraically closed field of arbitrary characteristic. The methods and techniques of Grothendieck, which have so changed the character of algebraic geometry in recent years, are used systematically throughout. Thus the classical material is presented from a new viewpoint. |

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### Contents

Raw Material on Curves on surfaces and | 1 |

The Fundamental Existence Problem and Two Analytic | 7 |

Re Representable Functors and Zariski | 26 |

Cartier Divisors | 61 |

199 | |

### Common terms and phrases

ample canonical Cartier divisor closed point closed subscheme closed subset codim coherent sheaf cohomology commutative construction corresponding curve D C F curve on F defined Definition degree dimension divisor class element equation equivalent exact sequence fact family of curves finite type flat follows functor h given group scheme hence Hilbert polynomial homomorphism hyperplane induced integral invertible sheaf irreducible isomorphism kernel Lecture 11 LEMMA Let f linear system locally free sheaf maximal ideal module Moreover multiplication noetherian scheme non-singular Num(F O-cycle O-divisor op(D open covering open sets Pic F power series pre-schemes prime ideal Proj projective space Proof Proposition prove quasi-coherent R-valued points result ring scheme S-valued sheaf of ideals sheaf on F sheaves spanned stalk Supp suppose surjective tensor theorem theory Topology truncation universal family vector space wn(X x(op Zariski tangent space zero