This eight-session seminar provides a foundational journey through the geometry of plane curves, exploring the intersection of polynomial algebra and geometric intuition. We will move from classical roots to the fundamental theorems that define the modern field.

Course Syllabus

• Session 1: Prehistory and Motivation
  Exploring classical origins, from Apollonian problems to the emergence of conic sections in antiquity.

• Session 2: Coordinate Geometry
  The transition from the affine to the projective plane. Defining curves as the zero locus of homogeneous polynomials.

• Session 3: Intersection Numbers
  Analyzing intersection multiplicity and the local behavior of curves at both singular and smooth points.

• Session 4: Bézout’s Theorem
  Study of the central result: why two projective curves of degrees n and m intersect in exactly nm points (counting multiplicity).

• Session 5: Geometry of Conics
  Classification and quadratic forms. Exploring the projective equivalence of curves of degree 2.

• Session 6: Cubics and Inflection Points
  Introduction to degree 3 curves, their inflection points, and a brief look into the group law.

• Session 7: Duality and Dual Curves
  The principle of projective duality: the correspondence between points and tangent lines.

• Session 8: Genus of a Plane Curve
  Concluding with the degree-genus formula and the connection between algebraic degree and the topology of the curve.

This minicourse is designed as a foundational gateway into the world of Projective Algebraic Geometry. Through the study of conics—the simplest yet most profound examples of algebraic curves—students will transition from classical Euclidean intuition to the powerful methods of modern geometry.

The primary objective is to equip students with the tools to classify conics across different settings: affine, projective, and over various algebraic fields. We will explore the deep interplay between linear algebra (bilinear forms and symmetric matrices) and geometry, culminating in an introduction to the Zaraki topology and the study of families of conics (pencils).

By the end of the course, participants will understand how these classical objects serve as a cornerstone for contemporary research in higher-dimensional algebraic varieties.

Key Learning Objectives:

• Mastering homogeneous coordinates and the structure of the projective plane.

• Understanding the transition from affine equations to projective varieties.

• Analyzing the parameter space of conics and its degenerations.

Organized by the DelTaMat project formed Jimmy Santamaria and Charlie Lozano at Universidad Mayor de San Andrés, this two-week course serves as an introduction to algebraic geometry through the study of affine algebraic curves. In the first week, we will explore the category of algebraic sets, laying the foundation for understanding their structure. The second week will focus on plane curves, examining their fundamental properties and key examples. 

The course is designed for students interested in developing a geometric intuition for algebraic varieties in a basic level.