We will develop the theory of algebraic varieties, which are zero sets of polynomial equations. We will start with some basic commutative algebra - define Groebner basis, chain conditions and talk about the ideal membership problem. Then we will discuss varieties - affine, projective, quasi-projective. We will prove Hilbert's Nullstellensatz (one of the most important theorems of classical algebraic geometry). We will talk about different notion of dimension and how those relate. We will define maps between varieites - morphisms, rational and birational maps. The remaining topics are (but not limited to) singularity theory, normalization and blow-ups, elimination theory, resultants, divisors on algebraic varieties. We will try to also focus on some computational aspects of algebraic geometry.
Office hours: Monday 5-6pm
|1||29 Aug - 1 Sept||Introduction, monomial orders, division algorithm, introducing the ideal membership problem|
|2||2 Sept - 8 Sept||Dickson's lemma, Hilbert's basis theorem, existence of Groebner basis, Noether's proposition, Buchberger's criterion|
|3||9 Sept - 15 Sept||Buchberger's algorithm, examples, Macaulay2, varieties|
|4||16 Sept - 22 Sept||Hilbert's Nullstellensatz||5||23 Sept - 29 Sept||Coordinate rings, dimension, morphisms||6||30 Sept - 6 Oct||Rational and birational maps, Projective varieties||7||7 Oct - 13 Oct||Quasi-projective varieties, products of quasi-projective varieties||8||14 Oct - 20 Oct||The image of projective morphism is closed||9||21 Oct - 28 Oct||Families of varieties, lines on surfaces||10||29 Oct - 3 Nov||Elimination theory, resultants||11||4 Nov - 10 Nov||Tangent space||12||11 Nov - 17 Nov||Resolution of singulatities overview, curves, normalization||13||24 Nov - 1 Dec||Blow-ups|