Lecturer : Thanasis Bouganis
Office : CM 126A
Ofiice Hours: Mondays 14:30 to 15:30 (or just write me an e-mail)
E-mail : athanasios.bouganis@durham.ac.uk
Term : Michaelmas 2013
Lectures :Literature
The lecture is based on the following two books. Although we will not follow them strictly, the material can be found in them and they may sometimes offer a different approach to the material.4H reading material: Affine Varieties and the Nullstellensatz (Chapter II, paragraph 3 of Miles Reid's "Undergraduate Algebraic Geometry")
Problem Sheets:
Problem Sheet 1 (due: Thursday Oct 24 at the beginning of the lecture)Problem Sheet 2 (due: Friday Nov 8, at 16:00, folder on my office door)
Problem Sheet 3 (due: Friday Nov 22, at 16:00, folder on my office door)
Problem Sheet 4 (due: Friday Dec 6, at 16:00, folder on my office door)
Content of Lectures
| Date | Content |
| Wednesday, 9 October 2013 (Week 1) | Introduction to algebraic geometry, definition of affine algberaic plane curves, Polynomial rings. |
| Thursday, 10 October 2013 (Week 1) | homogeneous polynomials in two variables, factorisation of homogeneous poynomials, irreducible components of curves, singularities. |
| Wednesday, 16 October 2013 (Week 2) | minimal polynomials, Study's Lemma (without proof), multiplicity of a point, tangent lines at a point. |
| Thursday, 17 October 2013 (Week 2) | Examples of points with multiplicities, geometric explanation of the tangent line at a non-singular point using the implicit function theorem, definition of projective spaces and basic properties. |
| Wednesday, 23 October 2013 (Week 3) | Properties of projective spaces, definition of complex projective curves, |
| Thursday, 24 October 2013 (Week 3) | notions of singular and non-singular points for projective curves, relation between affine and projective curves. |
| Wednesday, 30 October 2013 (Week 4) | irreducibility and singularity for projective and affine curves, Euler's Relation, examples of singular projective curves with an non-singular affine part. |
| Thursday, 31 October 2013 (Week 4) | Exercise Class. |
| Wednesday, 6 November 2013 (Week 5) | Bezout's Theorem (statement), definition of the resultant of two polynomials, properties of resultants. |
| Thursday, 7 November 2013 (Week 5) | properties of resultants (ctd), the degree of the resultant, other forms of the resultant |
| Wednesday, 13 November 2013 (Week 6) | other forms of the resultant (ctd), the notion of the projective transformation, applications of resultants: non-emptyness of the intersection of two projective curves . |
| Thursday, 14 November 2013 (Week 6) | the Weak Bezout's Theorem, finiteness of the singular locus of an irreducible projective curve |
| Wednesday, 20 November 2013 (Week 7) | the notion of intersection multiplicity, defining properties of the intersection multiplicity, existence of the intersection multiplicity. |
| Thursday, 21 November 2013 (Week 7) | uniqueness of the intersection multiplicity, |
| Wednesday, 27 November 2013 (Week 8) | uniqueness of the intersection multiplicity (ctd), proof of Bezout's Theorem, singularities and multiplicities |
| Thursday, 28 November 2013 (Week 8) | Exercise Class. |
| Wednesday, 04 December 2013 (Week 9) | Singularities and intersection multiplicities (ctd), intersection multiplicities equal to one and examples, introduction to cubics, the Hessian of a polynomial and points of inflection |
| Thursday, 05 December 2013 (Week 9) | Properties of the Hessian of a polynomial, existence of and bounds on the number of points of inflection of a non-singular projective curve, classification of projective non-singular cubic curves |
| Wednesday, 11 December 2013 (Week 10) | classification of projective non-singular cubic curves (ctd), intersections of lines and non-singular cubics |
| Thursday, 12 December 2013 (Week 10) | the abelian group structure on points of non-singular projective cubics |