This is a gentle introduction to Algebraic Geometry.
After studying introductory examples, we give a preliminary statement of
Bézout's Theorem which will serve as our main motivation for developing parts
of the theory of affine and projective varieties. Equipped with this, we will state
Bézout's theorem for projective plane curves and study some of its corollaries.
If time permits, we will move on to further topics such as birational maps and function fields,
resolution of singularities for curves and divisors on curves and the Riemann--Roch theorem.
- Introductory examples: linear subspaces, finite sets, conics, cubics, rational curves
- The category of affine varieties: affine algebraic sets, Zariski topology, irreducible decomposition, affine Nullstellensatz, affine coordinate rings, regular morphisms, algebraic groups
- The category of projective varieties: projective algebraic sets, projective Nullstellensatz, homogeneous coordinate rings, affine open coverings, Grassmannians
- Local properties of varieties: rational maps, local rings, tangent lines, multiple points, intersection numbers, the 27 lines on a nonsingular cubic surface
- Bézout's theorem for projective plane curves and some corollaries: Max Noether's fundamental theorem, Pascal's theorem, Pappus' theorem, addition on a cubic
Possible further topics:
- Birational maps and function fields
- Resolution of singularities for curves
- Divisors on curves and the Riemann--Roch theorem
- Additional topics based on the class's interest