Content for around two semesters worth of content on algebraic geometry. The first half of the course focuses on what is considered classical algebraic geometry, that is, chapter one of Harsthorne's book. The second half is somewhat more tricky, focusing on chapter two of Harsthorne, which is the topic of schemes. The key resource for classical algebraic geometry is Harsthorne (chapter one) and Algebraic Geometry by Andreas Gathmann (chapter 1-10). As for schemes, the outstanding book Introduction to Schemes by Geir Ellingsrud and John Christian Ottem provides all the details neccessary and more, from a very intuitive background.
Definitions, prime ideals, the Zariski topology, the Nullstellensatz correspondence, and coordinate rings as function rings.
Projective space VIA gluing affine patches, homogeneous idea,s, projective Nullstellensatz, and projective closures.
Zariski open and closed sets, irreducibility, constructible sets, and the intersection of topology with algebra in a variety.
Morphisms of varities, regular maps, local rings, a surface level understanding of sheaves.
Rational maps, function fields, dominant maps, birational equivalence, categories, and dualities.
Motivation for blowing-up, examples, resolving singularities, and ill-defined maps.
Tangent spaces, regular local rings, non-singular points, and the relation to smooth manifolds.
Dimension theory, chains of irreducible closed sets, and transcendence degree.
Product varieties, fibered products, parameter spaces, and moduli spaces.
Why bother with schemes? Limitations of classical varieties, gluing spectra, and a teaser for Grothenieck's vision of algebraic geometry.
Formal motivation, gluing affine schemes, and some philosophy
📘 Download Lecture Notes (PDF)Prime spectra, structure sheaf, basic open sets, and localisation.
📘 Download Lecture Notes (PDF)Pre-sheaves and sheaves, stalks, and morphisms of locally ringed spaces.
📘 Download Lecture Notes (PDF)Affine schemes, gluing VIA sheaves, open affine coverings, and examples.
📘 Download Lecture Notes (PDF)Morphisms, base change, open/closed immersions, and affine morphisms.
📘 Download Lecture Notes (PDF)Pullbacks of schemes, fibered products, and functoriality.
📘 Download Lecture Notes (PDF)Gluing spectra, constructing projective space VIA Proj, and comparison with classical varieties.
Ideal sheaves, closed immersions, fiber of morphisms, and universal properties.
Dimension of schemes, chains of prime ideals, and local dimension.
Reduced schemes, irreducibility, connectedness, associated points, and rounding up of topics.