Affine and projective spaces, affine and projective maps, Segre and Veronese embeddings, Desargues’s Theorem, Pappus’s Theorem, cross-ratio, projective duality

Plane curves: rational curves, linear systems of curves, conics and the Butterfly Theorem, Pascal’s Theorem, Chasles’s Theorem, the group structure on a planar cubic, Bezout’s Theorem

Affine algebraic varieties: Hilbert’s Basis Theorem, Zariski topology, irreducible components, Hilbert’s Nullstellensatz, the correspondence between the ideals and the algebraic sets, morphisms and rational maps between affine algebraic varieties

Projective varieties: graded rings and homogeneous ideals, the projective correspondence, morphisms, blow-ups, birational equivalence and rational varieties, Grassmannians

The basics of dimension theory

The basics of smoothness

Cubic surfaces and 27 lines. If time permits, other topics will be discussed such as abstract algebraic varieties, Chevaley’s Theorem, Riemann-Roch Theorem and its applications.