Math 763 -- Algebraic Geometry I -- Detailed list of topics
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Here is a detailed list of topics that I have covered in the past, based on 29 75-minute-classes. (To tell the truth, it seems very optimistic now... but who knows?)
- What is AG? Algebraic sets vs ideals. Statement of the Nullstellensatz.
- Proof of the Nullstellensatz. Zariski topology on $k^n$.
- Regular functions and regular maps. Coordinate rings.
- Zariski topology on algebraic varieties. Noetherian topological spaces. Principal open sets.
- Locally defined regular functions. Regularity of a function is local.
- Subvarieties of ${\mathbb A}^n$.
- Abstract algebraic varieties. Separated varieties.
- Subvarieties and products of varieties.
- Rational functions and rational maps.
- Dimension.
- Dimension of hypersurface.
- Complete intersections. Dimensions of fibers. ${\mathbb P}^n$.
- Projective varieties. Projective Nullstellensatz.
- Projective varieties are complete. Segre embedding.
- Grassmannians. Incidence variety.
- Dimension of fibers of projective maps.
- Chevalley's Theorem. Tangent space.
- Differential of a map. Smoothness. Local parameters.
- Taylor decomposition at a smooth point. Completed local ring.
- Regular local ring is a UFD.
- Smooth subvariety is lci. Birational vs biregular classification.
- Blow-ups.
- Resolution of singularities. Castelnuovo's criterion. Minimal surfaces.
- Divisors on smooth varieties. Weil divisors vs Cartier divisors.
- Pricipal divisors and the Picard group.
- Divisors on an affine variety as (fractional) ideals. Divisor classes as invertible modules.
- Algebraic vector bundles and line bundles.
- Linear systems and sections of line bundles.
- The Riemann-Roch Theorem.