Tutorial 3

• Polynomial methods : dimension counting,

• Identity theorem over finite fields,

• Geometry over finite fields

Tutorial 4

• Projective spaces and their linear subspaces

• Homogenization and points at infinity

• Projective vanishing lemma for Kakeya sets

Tutorial 5

• Conics 

• Projective transformations

• Quadratic forms

Tutorial 6

• Cubic curves 

• Points of inflections

• Elliptic curves and the group law

Tutorial 7

• More on projective varieties

• Segre embeddings

• 27  lines on cubic surface : example

Tutorial 8

• Reviews on commutative algebra : radical ideals

• Hilbert Basis Theorem,

• Hilbert Nullstelensatz Theorem

Tutorial 9

• Proof of Bezout's theorem.

Tutorial 10

• Hilbert functions

• Hilbert polynomials for projective varieties

• Hilbert polynomials for irreducible hypersurfaces