Topics in Algebraic Geometry: Introduction to the Theory of Schemes

A gentle introduction to the theory of schemes, emphasizing examples (continuing the course from Spring 2023.) We assume familiarity with quasi-projective varieties, maps between quasi-projective varieties, some basic awareness about manifolds and their homology, the language of commutative algebra, sheaves, locally ringed spaces, Spec and Proj, quasicoherent and coherent sheaves, and base change. Time permitting, we formulate the main results of intersection theory and discuss their applications. The primary references are Vakil's note 'The Rising Sea' and the book of Hartshorne 'Algebraic geometry'.