Title of the course: The Geometry of Shapes: An Introduction to Algebraic Surfaces
Instructor: Dr. Özgur İnce
Institution: Sivas Cumhuriyet University
Dates: 14-20 September 2026
Prerequisites: Complex Analysis
Level: Graduate, advanced undergraduate
Abstract: Complex algebraic surfaces are the zero sets of polynomials in two complex variables. However, this course is more than just solving equations; it is about understanding the shapes created by these equations. How many straight lines can lie on a surface? What happens when we ‘blow up’ a surface at a point? How did mathematicians organize this vast world of surfaces into a structured ‘library’? In this course, we will move away from abstract proofs and focus on visualizable examples—such as Cubic surfaces and K3 surfaces—to explore the fascinating geometry of algebraic surfaces.
Syllabus:
Day 1: From the Projective Plane to Surfaces: Understanding projective spaces including “points at infinity.” Defining the degree of a surface. The geometric definition of smooth surfaces vs. singular points.
Day 2: Curves and the Logic of Intersection: Viewing curves on a surface (divisors) as sub-geometries.The intersection number of two curves (C.D) and Bézout’s Theorem. What does “self-intersection” mean? Moving a curve on a surface.
Day 3: The Blow-up Operation The geometry of a blow-up: Replacing a point with a line. The art of separating two intersecting lines via blow-ups. The concept of Minimal Surfaces: Surfaces in their “simplest” form with no unnecessary blow-ups.
Day 4: Fingerprints of a Surface (Invariants): The genus of a surface and the intuitive “number of holes.”
Day 5: Famous Examples and the 27 Lines
Day 6: The Kodaira Classification (The Big Picture)
Language: TR, EN
Textbook:
– Miles Reid – Undergraduate Algebraic Geometry
– Andreas Gathmann – Algebraic Geometry Class Notes (https://agag-gathmann.math.rptu.de/class/alggeom-2021/alggeom-2021.pdf)
– Arnaud Beauville – Complex Algebraic Surfaces