School of Mathematics, Trinity College

Course 2318 - Elementary Projective and Algebraic Geometry 2011 (click for more information)

SF & JS Mathematics

Lecturer Dmitri Zaitsev

Examinations 2011: Credit will be given for the best 4 questions out of total 5 questions (each with 3 sub-questions). The theoretical questions will be within the scope of the current course and the practical problems within the scope of the current homework. The exam will count for 100% of the grade.

Problem Sheets in PDF: Sheet 1 Sheet 2 Sheet 3

Course outline:

Algebraic curves. Conics (or quadrics), their euclidean and affine classification over the fields of real and complex numbers. Projective plane, homogeneous coordinates, projective transformations. Lines in projective plane. Projective classification of conics. Parametrization of nondegenerate conics.

Homogeneous polynomials or forms. Roots of polynomials and their multiplicities. Bézout's Theorem, proof when one of the curve is a line or a quadric. Factorization of forms vanishing along lines and nondegenerate conics. Spaces of d-forms vanishing at certain points and their dimensions. Applications to quadrics passing through 5 points and cubics passing through 9 points. Pascal's Theorem.

Nodal and cuspidal cubics, their parametrization. Tangent lines. Group law on a cubic.

Affine algebraic sets, their ideals. Noetherian rings. Hilbert Basis Theorem. Algebraic sets defined by ideals, their properties. Zariski topology. Termination of descending chains of algebraic sets. Irreducible algebraic sets, their relation with prime ideals. Unique decomposition of algebraic set into irreducible components.

Nullstellensatz (Hilber Zero Theorem) and Weak Nullstellensatz. Proof of the Nullstellensatz assuming Weak Nullstellensatz.

Polynomial functions on affine algebraic sets. Coordinate ring. Polynomials maps between affine algebraic sets. Relation between polynomial maps and coordinate ring homomorphisms. Polynomial isomorphisms. Affine varieties. Rational functions on affine algebraic sets. Regular points of rational functions. Rational maps. Dominant maps.

Projective algebraic sets. Homogeneous ideals and correspondence between them and projective algebraic sets. The affine cone over a projective algebraic set. Rational functions on projective algebraic sets, rational maps between them. Regular points of rational functions and maps. Morphisms and isomorphisms. Segre embedding of the product of two projective spaces into another projective space. Finite unions, finite products, and arbitrary intersections of projective algebraic sets are again projective algebraic.

M. Reid, Undergraduate Algebraic Geometry, London Mathematical Society Student Texts, Cambridge University Press.
I.R. Shafarevich, Basic algebraic geometry, Springer, 1994.
I. Dolgacev, Introduction to Algebraic Geometry, Lecture Notes.
D. Mumford, The Red Book of Varieties and Schemes, Lecture Notes in Mathematics.

Some links.
Lecture courses on video by Miles Reid
Wolfram Mathworld Pages on Algebraic Geometry
Wikipedia Pages on Algebraic Geometry

Further Online Lecture Notes.
Elementary Algebraic Geometry by David Marker

Old courses homepages:
Course 2318 - Algebraic Geometry 2009-10 by Calin Lazaroiu

For exam-related problems look in TCD past examination papers and Mathematics department examination papers.

Student Counselling Service

I will appreciate any (also critical) suggestions that you may have for the course. Let me know your opinion, what can/should be improved, avoided etc. and I will do my best to follow them. Feel free to come and see me if and when you have a question about anything in this course. Or use the feedback form from where you can also send me anonymous messages.