## An Introduction to Invariants and ModuliIncorporated in this volume are the first two books in Mukai's series on Moduli Theory. The notion of a moduli space is central to geometry. However, its influence is not confined there; for example, the theory of moduli spaces is a crucial ingredient in the proof of Fermat's last theorem. Researchers and graduate students working in areas ranging from Donaldson or Seiberg-Witten invariants to more concrete problems such as vector bundles on curves will find this to be a valuable resource. Among other things this volume includes an improved presentation of the classical foundations of invariant theory that, in addition to geometers, would be useful to those studying representation theory. This translation gives an accurate account of Mukai's influential Japanese texts. |

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### Contents

Invariants and moduli | 1 |

Rings and polynomials | 51 |

Algebraic varieties | 77 |

Algebraic groups and rings of invariants | 116 |

The construction of quotient varieties | 158 |

The projective quotient | 181 |

The numerical criterion and some applications | 211 |

Grassmannians and vector bundles | 234 |

Curves and their Jacobians | 287 |

Stable vector bundles on curves | 348 |

Moduli functors | 398 |

Intersection numbers and the Verlinde formula | 437 |

487 | |

495 | |

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### Common terms and phrases

action affine quotient affine space affine variety algebraic group algebraic variety called Chapter coefficients cohomology consider construction Corollary cubic defined Definition degree denote determines dimension element elementary sheaf equation equivalent exact sequence Example exists field of fractions finitely free module function field functor G-invariant genus Gieseker point GL(N global sections gluing Grassmannian hence Hilbert series homogeneous polynomial homomorphism hypersurfaces integral domain isomorphic Jacobian Lemma line bundle line subbundle linear map linearly reductive locally free matrix maximal ideal moduli space monomials morphism nonsingular nonzero open set orbit parametrised point p e polynomial ring Proj quotient quartic quasiparabolic quotient variety rank 2 vector rational function representation ring of invariants satisfying semiinvariant semistable skew-symmetric SL(n stable subbundle subring subset subspace Suppose surjective tangent tensor Theorem valuation ring vector bundle vector space zero