Math
606: Introduction to Algebraic Geometry
4115 MATH Building, x5-5080, atma@math
Class timings: MWF noon-1pm in MATH 0407
Office hours: TBA
This course aims to provide an introduction to basic algebraic
geometry (BAG). There are several
approaches to AG. We shall try to use the complex analytic approach at the
beginning and then (mid-way) switch to the algebraic approach. Our main
emphasis will be on algebraic curves (and later, perhaps their moduli), for
these illustrate very clearly the fundamental role of algebraic geometry in all
of mathematics.
Syllabus:
Part I:
Compact complex manifolds, cohomology, Hodge Theory, Projective algebraic
varieties, Chow’s theorem, Kodaira’s theorems, Hodge Index Theorem,
Hodge-Riemann bilinear relations, Lefschetz theorems, complex tori vs abelian
varieties, Hodge conjecture.
Part II:
Compact Riemann surfaces = (smooth projective) complex algebraic curves, Galois
groups and fundamental groups, geometry of algebraic curves in projective space
(singular points, inflection points, Bezout theorem, etc) divisors, line
bundles, Jacobian, sheaves and their cohomology with emphasis on algebraic
curves (Riemann-Roch theorem and its applications).
Part III:
The algebraic theory of varieties and schemes (Hilbert nullstellensatz, locally
ringed spaces, structure sheaf, smooth varieties, rational maps, morphisms,
blowup, resolution of singularities, divisors and embeddings into projective
space, canonical class, Serre duality, Kodaira vanishing, vector bundles,
locally free sheaves, etc). Varieties over finite fields and the Weil
conjectures.
Part IV:
Representable functors, algebraic groups, Grassmannians, Chow varieties, moduli
spaces, moduli of vector bundles on curves, moduli of curves, arithmetic schemes
(Spec Z).
(Perhaps
this is way too optimistic!)
Textbooks:
Principles
of Algebraic Geometry by P. Griffiths and J. Harris (Wiley Classics)
Hodge Theory
and Complex Algebraic Geometry I by C. Voisin (Cambridge Studies in Advanced
Math)
Algebraic
Geometry by R. Hartshorne (GTM 52, Springer)
Invitation
to Algebraic Geometry by K. Smith et al (Springer)
Other recommended books:
Basic
algebraic geometry I, II by I. R. Shafarevich (Springer)
Algebraic
geometry I: complex projective varieties by D. Mumford (Springer)
Curves and
their Jacobians by D. Mumford (Springer)
The red book
of varieties and schemes by D. Mumford
Complex
algebraic curves by F. Kirwan (Cambridge University)
Quelques
aspects de la surfaces de Riemann by E. Reysatt (Birkhauser)
From number
theory to physics edited by M.
Waldschmidt (articles by J.-B. Bost, etc) (Springer)
Algebraic
varieties by G. Kempf (London Math Soc)
Algebraic
geometry: introduction to schemes by I.G. Macdonald (Benjamin)
Introduction
to commutative algebra and algebraic geometry by E. Kunz (Birkhauser)
Geometry of
schemes by D. Eisenbud and J. Harris (GTM)
Algebraic
geometry: a first course by J. Harris (GTM)
Undergraduate
algebraic geometry by M. Reid (London Math Soc Student Texts 12)
Fundamental
algebraic geometry: Grothendieck’s FGA Explained by L. Gottsche et al (AMS)
Methods of
algebraic geometry by W. Hodge and D. Pedoe (Cambridge Math Library)
Algebraic
geometry 1,2,3 by K. Ueno (AMS)
Advances in
moduli theory by K. Ueno (AMS)
Algebraic
curves: an introduction to algebraic geometry by W. Fulton (Addison-Wesley)
Algebraic
geometry I-V edited by Parshin and Shafarevich (Encyclopedia of Math Sciences,
Springer)
Algebraic
geometry-Arcata 1974 edited by R. Hartshorne (AMS PSPUM)
Homeworks:
Problems will be assigned regularly, but not collected!
Schedule: ??