Math 606: Introduction to Algebraic Geometry

Fall 2006

 

Niranjan Ramachandran

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, Chowí»s theorem, Kodairaí»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: Grothendieckí»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: ??