## Lectures on Curves on an Algebraic Surface |

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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