Spring 2009

## Syllabus

The following is a tentative syllabus for Math 189, Spring 2009. The actual topics covered will depend upon interest and pace.
The rough outline of the course is as follows. We will follow Reid in more or less linear fashion for a good part of the semester. This includes an introduction to projective geometry, plane conics, cubics and general curves, an introduction to affine varieties, the Nullstellensatz, rational functions, morphisms, birational maps, projective varieties, and a discussion of tangent spaces and singularities. The last topic in Reid is the fun enumerative example of 27 lines of a cubic surface. Depending on pace and interest, we may discuss other topics such as syzygies and Schubert calculus. What follows is a terse outline of what has been (or will be) discussed in each lecture.

## Lecture topics

1. Introduction to projective space via enumerative geometry. Introduction to curves in the plane. Unique conic through 5 general points in complex projective plane.
2. Conics in R^2 and affine transformations. Conics in real projective plane and projective transformations. Bilinear symmetric forms and quadratic forms. Definition of nondegeneracy.
3. Parametrized lines and conics. Homogenization and associated non-homogeneous polynomials. Roots of forms and homogeneous version of fundamental theorem of algebra. Multiplicity.
4. Bezout's theorem (special cases). Moduli of conics through specified points in real projective plane. Pencils of conics.
5. Definition of affine and projective variety. Distinguished (open) subsets of projective space and description of projective varieties in terms of affine patches. Projectivizations of vector spaces (without specified basis). Examples of projective varieties: lines, hyperplanes and general linear subspaces and the twisted cubic.
6. Examples of varieties continued: the rational normal curve. Main lecture topic: plane cubics. Examples in affine space, nodal and cuspidal cubic. Plane cubics not admitting rational parametrization. Linear system and intersections with points.
7. Main topic: group law on cubics. Linear systems continued. Dimension of space of cubic forms through collections of points. A first glance at tangent lines. Group law on cubics. Normal form of cubic equation and simplified group law.
8. Group law continued. Definition of irreducible conic. Relationship between unions of varieties and products of functions. Associativity of the group law. The mystic hexagon.
9. Main topics: the V correspondence, introduction to topology. Definition of the correspondence V. Definition of algebraic set. Refresher on Noetherian rings and Hilbert Basis Theorem. Algebraic sets form the closed sets of a topology on affine space. Definition of topological space. Examples: the real line, finite sets. Definition of irreducible algebraic set.
10. Main topic: Nullstellensatz. The I correspondence. Are V-I inverse maps? Example: the ideal in k[x,y] generated by x^2. Definition of radical ideal. Characterization of maximal ideals of polynomial rings. Proof of weak Nullstellensatz. Proof of Nullstellensatz via trick of Rabinowitsch.
11. Main topics: Nullstellensatz, functions on varieties. Detailed review of proof of Nullstellensatz using weak version. Bijection between radical ideals and algebraic subsets of affine space. Example of varieties defined via ideals. Definition of coordinate ring of affine variety and functions on a variety.
12. Definition of polynomial map between varieties. Definition of isomorphism of affine varieties. Bijection between polynomial maps and homomorphisms of coordinate rings of affine varieties.
13. Definition of function field, rational function, regular values and the domain of a rational function. The local ring of a variety at a point. The domain is open and dense. Regular rational functions are polynomial functions.
14. Rational maps between varieties. Dominant maps. Bijections between dominant rational maps and homomorphisms of function fields of affine varieties. Morphisms on (open subsets of) affine varieties. Standard open sets and their properties.
15. Examples of bijections between maps between varieties and homomorphisms of rings. A first look at the blowup of the affine plane at the origin. Homogeneous ideals of the polynomial ring. The projective I-V correspondence. The Zariski topology on projective space. The affine cone over a projective variety. The projective Nullstellensatz.