Instructor: Pal ZSAMBOKI PhD

Text:  Class notes will be available online.
Our main reference is William Fulton: Algebraic curves, An introduction to algebraic geometry available on the author's website .

The following textbooks are of a similar level and thus will serve as our main sources for further examples.
— Miles Reid: Undergraduate algebraic geometry available on the author's website
— Joe Harris: Algebraic geometry, A first course
— Gerd Fischer: Plane algebraic curves

Prerequisites: The main prerequisite is experience with manipulation of polynomials in several variables with coefficients in fields, and familiarity with basic ring-theoretic concepts: you can find a reminder about these in Section 1.1 of Fulton: Algebraic curves. We will study the Zariski topology on algebraic sets. Prior exposure to abstract topological spaces and open/closed sets might help; in any case, these will be introduced in class. Similarly, I will at times emphasize the category theoretical point of view as I find that to be the clearest way of seeing how algebraic and geometric notions correspond. Therefore, basic familiarity of categories, functors, equivalences and universal properties might help appreciate this point of view; these notions will be introduced in class as well.

Description: 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.

Topics:

1. Introductory examples: linear subspaces, finite sets, conics, cubics, rational curves
2. The category of affine varieties: affine algebraic sets, Zariski topology, irreducible decomposition, affine Nullstellensatz, affine coordinate rings, regular morphisms, algebraic groups
3. The category of projective varieties: projective algebraic sets, projective Nullstellensatz, homogeneous coordinate rings, affine open coverings, Grassmannians
4. Local properties of varieties: rational maps, local rings, tangent lines, multiple points, intersection numbers, the 27 lines on a nonsingular cubic surface
5. 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