## The Moduli Space of CurvesRobert H. Dijkgraaf, Carel Faber, Gerard B.M. van der Geer This generalization of geometry is bound to have wide spread repercussions for mathematics as well as physics. The unearthing of it will entail a new golden age in the interaction of mathematics and physics. E. Witten (1986) The idea that the moduli space Mg of curves of fixed genus 9 - that is, the algebraic variety that parametrizes all curves of genus 9 - is an intriguing object in its own right seems to have come slowly. Although the para meters or moduli of curves surface in Riemann's famous memoir on abelian functions (from 1857) and in work of Hurwitz and later were considered by the geometers of the Italian school, for a long time they attracted attention only in the special case 9 = 1, where they were studied in the framework of the theory of modular functions. The work of Grothendieck, who in the early sixties pointed the way towards the right approach, and the subsequent construction (in 1965) of the moduli space Mg by Mumford were the first foundational work, to be followed by the construction of a compactification Mg by Deligne and Mumford in 1969. The theorem of Harris and Mumford saying that for 9 sufficiently large the space Mg is of general type was the first big insight in its structure. |

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### Contents

CONTENTS | 34 |

Quantum cohomology of rational surfaces | 35 |

surfaces | 55 |

Associativity in general | 68 |

Enumerative consequences of associativity | 74 |

Quantum intersection rings | 81 |

Mirror symmetry and elliptic curves | 149 |

R Dijkgraaf | 162 |

Operads and moduli spaces of genus 0 Riemann surfaces | 199 |

Resolution of diagonals and moduli spaces | 231 |

Ginzburg | 266 |

The cohomology of algebras over moduli spaces | 305 |

Enumeration of rational curves via torus actions | 335 |

Kontsevich | 368 |

INTRODUCTION | 480 |

### Other editions - View all

The Moduli Space of Curves Robert H Dijkgraaf,Carel Faber,Gerard B M Van Der Geer No preview available - 1995 |

The Moduli Space of Curves Robert H. Dijkgraaf,Carel Faber,Gerard B.M. van der Geer No preview available - 2011 |

The Moduli Space of Curves Robert H. Dijkgraaf,Carel Faber,Gerard B.M. van der Geer No preview available - 1995 |

### Common terms and phrases

action algebraic stack arithmetic genus Aſc associated automorphism canonical Chow ring codimension cohomology ring complex component compute configuration conformal field theory contains corresponding cubic curves of genus cyclic points defined deformation degree Del Pezzo surface denote diagonal differential dimension divisor classes double points dual edge element equation equivalent equivariant fiber field theory finite group formula function fundamental group genus g geometry graph Hence holomorphic homotopy intersection invariant irreducible isomorphism Kontsevich Lemma Lie algebra lines locus manifold Math modular moduli space Mošn morphism node notation obtained operad parametrizing permutation plane polynomial projective proof Proposition quadrics quantum cohomology quantum product quartic quotient ramification points rational curves relations relevant class S-module sheaf singular smooth point stable curves structure subset symmetric tautological classes Theorem topological tree trivial vector bundle vector space vertex vertices