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.