MATH 622- COMPLEX ALGEBRAIC GEOMETRY
			   Ron Donagi
			   Fall  1999
		
This page lists the topics covered in class as well as the homework problems.

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Lecture 1: 9/14

Introduction
Affine varieties
Complex and Zariski topologies
Irreducible vs. connected
Projective space, projective varieties
Examples of affine and projective varieties

Homework 1 (due Thursday, 9/23): 
Show that any two (irreducible) curves, with the Zariski topology, are 
homeomorphic.

Homework 2 (due Thursday, 9/23): 
Classify quadrics (=hypersurfaces of degree 2) in CP^n,RP^n,CA^n,RA^n, 
up to automorphisms of the ambient projective or affine space.


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Lecture 2: 9/16

Presheaves
Sheaves
Ringed spaces
Manifolds 
Varieties
Schemes
Vector bundles, fiber bundles

The point of this lecture is that a variety is something which locally 
looks like an affine variety, in the same sense that a manifold is 
something which locally looks like an open subset of a vector space. 

Homework 3 (due Thursday, 9/23): 
Is the cokernel of a sheaf morphism necessarily a sheaf?

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Lecture 3: 9/21

More projective varieties
Quadrics
Projective duality
The dual variety


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Lecture 4: 9/23

Dual varieties
Parameter spaces: definition
Examples: the parameter spaces of points, pairs of points, lines, 
conics, singular conics.
The Veronese variety

Homework 4 (due Tuesday, 10/5): 
The parameter space of all plane cubic curves can be identified with the 
projective space P^9. This has a natural stratification by singularity 
type of the cubic. (This is analogous to the stratification of the 
parameter space of conics (=P^5) by the singularity type of the conic, 
or equivalently by the rank of the corresponding symmetric matrix.)

(a)Identify nine subsets of P^9, each parametrizing cubics with a 
particular singularity type. (E.g.: there is an open stratum of 
non-singular cubics, a codimension 1 stratum of cubics which intersect 
themselves once transversally, a codim 2 locus of cubics with a 
cusp,...)

(b)Find the dimension of each stratum.

(c)Figure out which strata are in the closure of which other strata. 

(d)Whenever possible, find the degree (as projective variety in P^9) of 
the closure of each stratum. (Some of these are hard.)

(e)What is the degree of the dual variety of a plane cubic, and what 
does it have to do with the rest of this problem.

(f)For each stratum, are all the curves parametrized by the points of 
that stratum isomorphic to each other (via automorphisms of P^2)?

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Lecture 5: 9/28

Properties of varieties: 
  dimension
  non-singularity
  compactness
  separatedness.
Constructions of varieties: 
  products 
  cones
  projections.

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Lecture 6: 9/30

Blowing up.
Topology of curves.
Hurwitz' formula.
Genus of a plane curve.
Pencils of plane curves, degree of the dual curve.
Linear systems of plane curves.

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Lecture 7: 10/5

More construction techniques: 
Projection of a variety from a point on it.
Grassmannians, quotients, symmetric products, Hilbert schemes.



References:
----------

[C]  Cox, Recent developments in toric geometry
alg-geom/9606016
[F]  Fulton, Toric varieties
[GH] Griffiths, Harris: Principles of algebraic geometry
[H]  Hartshorne, Algebraic geometry (GTM 52)
[M1] Mumford, Algebraic geometry I: complex projective varieties 
      (Grundlehren 221)
[M2] Mumford, The red book
[M3] Mumford, Curves and their Jacobians
[O]  Oda, Convex bodies and algebraic geometry (Ergebnisse 15)
[Wa] Warner, Foundations of differentiable manifolds and Lie groups 
      (GTM 94)
[We] Wells, Differential analysis on complex manifolds (GTM 65)