An Introduction to Invariants and Moduli

✍ Scribed by Shigeru Mukai

• Publisher: Cambridge University Press
• Year: 2003
• Tongue: English
• Leaves: 524
• Edition: 1
• Category: Library

✦ Synopsis

Incorporated 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.

✦ Table of Contents

Cover
About
Cambridge Studies in Advanced Mathematics
Title Page
Copyright Page
Contents
Preface
Acknowledgements
Introduction
(a) What is a moduli space?
(b) Algebraic varieties and quotients of algebraic varieties
(c) Moduli of bundles on a curve
Chapter 1. Invariants and moduli
1.1 A parameter space for plane conics
1.2 Invariants of groups
(a) Hilbert series
(b) Molien's formula
(c) Polyhedral groups
1.3 Classical binary invariants
(a) Resultants and discriminants
(b) Binary quartics
1.4 Plane curves
(a) Affine plane curves
(b) Projective plane curves
1.5 Period parallelograms and cubic curves
(a) Invariants of a lattice
(b) The Weierstrass ℘ function
(c) The ℘ function and cubic curves
Exercises
Chapter 2. Rings and polynomials
2.1 Hilbert's Basis Theorem
2.2 Unique factorisation rings
2.3 Finitely generated rings
2.4 Valuation rings
(a) Power series rings
(b) Valuation rings
2.5 A diversion: rings of invariants which are not finitely generated
(a) Graded rings
(b) Nagata's trick
(c) An application of Liouville's Theorem
Exercises
Chapter 3. Algebraic varieties
3.1 Affine varieties
(a) Affine space
(b) The spectrum
(c) Some important notions
Morphisms
Products
General spectra and nilpotents
Dominant morphisms
Open immersions
Local properties
3.2 Algebraic varieties
(a) Gluing affine varieties
(b) Projective varieties
3.3 Functors and algebraic groups
(a) A variety as a functor from algebras to sets
(b) Algebraic groups
3.4 Completeness and toric varieties
(a) Complete varieties
(b) Toric varieties
(c) Approximation of valuations
Exercises
Chapter 4. Algebraic groups and rings of invariants
4.1 Representations of algebraic groups
4.2 Algebraic groups and their Lie spaces
(a) Local distributions
(b) The distribution algebra
(c) The Casimir operator
4.3 Hilbert's Theorem
(a) Linear reductivity
(b) Finite generation
4.4 The Cayley–Sylvester Counting Theorem
(a) SL(2)
(b) The dimension formula for SL(2)
(c) A digression: Weyl measure
(d) The Cayley–Sylvester Formula
(e) Some computational examples
4.5 Geometric reductivity of SL(2)
Exercises
Chapter 5. The construction of quotient varieties
5.1 Affine quotients
(a) Separation of orbits
(b) Surjectivity of the affine quotient map
(c) Stability
5.2 Classical invariants and the moduli of smooth hypersurfaces in P^n
(a) Classical invariants and discriminants
(b) Stability of smooth hypersurfaces
(c) A moduli space for hypersurfaces in P^n
(d) Nullforms and the projective quotient map
Exercises
Chapter 6. The projective quotient
6.1 Extending the idea of a quotient: from values to ratios
(a) The projective spectrum
(b) The Proj quotient
(c) The Proj quotient by a GL(n)-action of ray type
6.2 Linearisation and Proj quotients
6.3 Moving quotients
(a) Flops
(b) Toric varieties as quotient varieties
(c) Moment maps
Chapter 7. The numerical criterion and some applications
7.1 The numerical criterion
(a) 1-parameter subgroups
(b) The proof
7.2 Examples and applications
(a) Stability of projective hypersurfaces
(b) Cubic surfaces
(c) Finite point sets in projective space
Chapter 8. Grassmannians and vector bundles
8.1 Grassmannians as quotient varieties
(a) Hilbert series
(b) Standard monomials and the ring of invariants
(c) Young tableaux and the Plücker relations
(d) Grassmannians as projective varieties
(e) A digression: the degree of the Grassmannian
8.2 Modules over a ring
(a) Localisation
(b) Local versus global
(c) Free modules
(d) Tensor products and flat modules
8.3 Locally free modules and flatness
(a) Locally free modules
(b) Exact sequences and flatness
8.4 The Picard group
(a) Algebraic number fields
(b) Two quadratic examples
8.5 Vector bundles
(a) Elementary sheaves of modules
(b) Line bundles and vector bundles
(c) The Grassmann functor
(d) The tangent space of the functor
Exercises
Chapter 9. Curves and their Jacobians
9.1 Riemann's inequality for an algebraic curve
(a) Prologue: gap values and the genus
(b) Divisors and the genus
(c) Divisor classes and vanishing index of speciality
9.2 Cohomology spaces and the genus
(a) Cousin's problem
(b) Finiteness of the genus
(c) Line bundles and their cohomology
(d) Generation by global sections
9.3 Nonsingularity of quotient spaces
(a) Differentials and differential modules
(b) Nonsingularity
(c) Free closed orbits
9.4 An algebraic variety with the Picard group as its set of points
(a) Some preliminaries
(b) The construction
(c) Tangent spaces and smoothness
9.5 Duality
(a) Dualising line bundles
(b) The canonical line bundle
(c) De Rham cohomology
9.6 The Jacobian as a complex manifold
(a) Compact Riemann surfaces
(b) The comparison theorem and the Jacobian
(c) Abel's Theorem
Exercises
Chapter 10. Stable vector bundles on curves
10.1 Some general theory
(a) Subbundles and quotient bundles
(b) The Riemann–Roch formula
(c) Indecomposable bundles and stable bundles
(d) Grothendieck's Theorem
(e) Extensions of vector bundles
10.2 Rank 2 vector bundles
(a) Maximal line subbundles
(b) Nonstable vector bundles
(c) Vector bundles on an elliptic curve
10.3 Stable bundles and Pfaffian semiinvariants
(a) Skew-symmetric matrices and Pfaffians
Even skew-symmetric matrices
Odd skew-symmetric matrices
Skew-symmetric matrices of rank 2
(b) Gieseker points
(c) Semistability of Gieseker points
10.4 An algebraic variety with SU C (2, L) as its set of points
(a) Tangent vectors and smoothness
(b) Proof of Theorem 10.1
(c) Remarks on higher rank vector bundles
Exercises
Chapter 11. Moduli functors
11.1 The Picard functor
(a) Fine moduli and coarse moduli
(b) Cohomology modules and direct images
(c) Families of line bundles and the Picard functor
(d) Poincaré line bundles
11.2 The moduli functor for vector bundles
(a) Rank 2 vector bundles of odd degree
(b) Irreducibility and rationality
(c) Rank 2 vector bundles of even degree
11.3 Examples
(a) The Jacobian of a plane quartic
(b) The affine Jacobian of a spectral curve
(c) The Jacobian of a curve of genus 1
(d) Vector bundles on a spectral curve
(e) Vector bundles on a curve of genus 2
Exercises
Chapter 12. Intersection numbers and the Verlinde formula
12.1 Sums of inverse powers of trigonometric functions
(a) Sine sums
(b) Variations
(c) Tangent numbers and secant numbers
12.2 Riemann–Roch theorems
(a) Some preliminaries
(b) Hirzebruch–Riemann–Roch
(c) Grothendieck–Riemann–Roch for curves
(d) Riemann–Roch with involution
12.3 The standard line bundle and the Mumford relations
(a) The standard line bundle
(b) The Newstead classes
(c) The Mumford relations
12.4 From the Mumford relations to the Verlinde formula
(a) Warming up: secant rings
(b) The proof of formulae (12.2) and (12.4)
12.5 An excursion: the Verlinde formula for quasiparabolic bundles
(a) Quasiparabolic vector bundles
(b) A proof of (12.6) using Riemann–Roch and the Mumford relations
(c) Birational geometry
Bibliography
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Index