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Kevin Coombes

At present, copies of the documents are only available as DVI files. If you're interested in seeing them and can only handle PostScript, then I'm sure something can be arranged.

Contents

• Notes on Algebraic Geometry
• Notes on Étale Cohomology
• Notes on Algebraic Surfaces
• Notes on Curves on Surfaces
• Infinite Voyage: An introduction to calculus
• Mathematical Hypertexts
• A History of the Idea of Functions

• Affine Varieties
• Rational Maps
• Local Structure of Varieties

• Flat Modules
• Descent Theory
• Ramification and Differentials
• Étale Morphisms
• Sites and Sheaves
• Points and Stalks
• Naturality
• Étale Cohomology

• Basic Tools
• Intersection Theory
• Chern Classes
• Numerical Invariants
• Ampleness and Adjunction
• Castelnuovo's Criterion
• Ruled Surfaces
• Kodaira Dimension Zero
• Abelian and Enriques Surfaces

• Preface
• Introduction
• Pointed Remarks
• Grassmann and the Functor of Points
• Hilbert Polynomials
• Algebraic K-theory and Algebraic Cycles
• Flatness
• Relative Effective Cartier Divisors
• Linear Systems
• Vanishing Theorems
• The Semicontinuity Theorem
• Universal Families of Curves
• The Picard Scheme
• Good Curves
• The Hodge Index Theorem
• Independent Zero Cycles

Beginning at the University of Michigan and continuing at the University of Maryland, I've spent time teaching honors calculus. At both institutions, there are two flavors of honors calculus. The highest level course is designed for talented freshman who are extremely likely to become mathematics majors. The second honors calculus course is designed for general honors students who want a more interesting introduction to calculus. I've accumulated a set of notes to supplement a class of the second kind.

Having struggled mightily on several occassions to convey the ideas of algebraic geometry to a class of graduate students, I've become convinced that there is no good linear way to present this material. I think it would be wonderful to have a set of hypertext notes in algebraic geometry. Ideally, these would allow you to move back and forth between examples, varieties, and schemes. They would allow multiple tables of contents to the same material, with my favorite being a list of major theorems. You could start with the statement of a theorem, and drill your way down to the underlying definitions and supporting examples, leaving the lemmas and the generalizations hidden away until you absolutely needed them.

As far as I know, no such set of notes exists. Had I an infinite number of hours in each day, I'd sit down to write them. But I don't, so I haven't. If you'd like a very small sample of how hypertext can change the presentation of mathematical proofs, you can take a look at my notes on prime numbers.

In any event, right after I find the time to write a hypertext treatment of algebraic geometry, I'd like to find the extra time to write a history of functions, starting with an explanation of why the Greeks didn't have them (Hint: their physics is all statics and no dynamics.) and continuing into such twentieth century developments as distributions and functors.

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