Advanced Topics in Algebraic Geometry

Outline of talk: My goal is to introduce a few more advanced topics in algebraic geometry but not to go into too much detail. This will be a survey of (1) Elimination Theory and review from last time (BRIEFLY) (a) History and goals (b) Geometric Extension Theorem (2) Invariant Theory (3) Dimension Theory (a) Krull Dimension (b) Hilbert Polynomial (c) Dimension of a variety (4) Syzygies (a) Definition (b) Free resolutions (c) Hilbert’s Syzygy Theorem (5) Intersection Theory (a) Bezout’s Theorem (b) Discriminant (c) Sylvester Matrix (d) Resultants (e) Grassmannians Recall basics of algebraic geometry: sets in affine space correspond to functions in k[x1, . . . , xn] which vanish on those sets. This is given by I → V (I) (an affine algebraic set) and by V → I(V ) (an ideal). It was inclusion-reversing and had other friendly properties like I(V (I)) = I. An affine variety was an affine algebraic set which could not be written as a union of two other non-trivial affine algebraic sets. Almost all the familiar algebraic geometry from affine space holds over to projective space but we deal with homogeneous polynomials (all monomials of the same degree) so that they will be welldefined on points. This is because for projective points (x0 : · · · : xn) is equivalent to (λx0 : · · · : λxn) for all λ ∈ k \ {0}. But for homogeneous polynomials this just corresponds to multiplication