| Day | Topics | Sections in IVA |
|---|---|---|
| Wed. 1/18 | Introduction: to algebraic geometry: classification, singularities, parametrization. |
. |
| Mon. 1/23 | Algebra: algebraically closed fields, polynomial rings, field of fractions. Geometry: varieties, their intersection and union. |
I.1,2 |
| Wed. 1/25 | Parametrization and Implicitization. | I.2,3. |
| Mon. 1/30 | Ideals in polynomial rings. | I.4. |
| Wed. 2/1 | Univariate polynomial rings versus
multivariate polynomial rings.
The Euclidean algorithm. |
I.5. |
| Mon. 2/6 | Monomial Orderings | II.1,2 |
| Fri. 2/10 | Division in multivariate polynomial rings. Division Algorithm in Magma. |
II.3 poly_div.mg |
| Wed. 2/15 | Dickson's Lemma. | II.4 |
| Fri. 2/17 | Hilbert basis theorem, Grobner basis. | II.5-8 |
| Wed. 2/22 | S-polynomials and Grobner basis.
Buchberger's algorithm. |
II.6,8 |
| Fri. 2/24 | Rings and modules, presentations and syzygys. | II.6,8,9 |
| Mon. 2/27 | The syzygy module for monomial ideals. Relation to reduction by G |
II.9 II.1, 2 |
| Wed. 3/1 | Improving Buchberger's algorithm. Elimination and extension. | III.1 |
| Mon. 3/6 | Elimination, extension and closure theorems. Sylvester matrix |
III.2, 5 |
| Wed. 3/8 | Sylvester matrix, resultants.
Proof of the Extension theorem. |
III.5,6 |
| Mon. 3/20 | Implicitization. | III.3 |
| Wed. 3/22 | Radical ideals and Hilbert's Nullstellensatz.
Zariski closure. |
IV.1,2 |
| Mon. 3/27 | The algebra-geometry dictionary. Sums, products and intersections of ideals. | IV.2,3. |
| Wed. 3/29 | Computation of the intersection of two ideals.
Proof of the closure theorem. |
IV.3,4. |
| Mon. 4/3 | Ideal quotients, their computation and the geometric interpretation. | IV.4. |
| Wed. 4/5 | Cancelled. | . |
| Mon. 4/10 | Irreducible varieties and minimal decomposition. | IV.5,6 |
| Wed. 4/12 | Ring homomorphisms and quotients. The coordinate ring of a variety. |
V.1-4 |
| Mon. 4/17 | The spectrum of a ring, the Zariski topology | V.3. IV.5 |
| Wed. 4/19 | Finite varieties. Polynomial and rational maps of varieties. | V.5 |
| Mon. 4/24 | Bezout's theorem as an introduction to algebraic geometry. | . |
| Wed. 4/26 | Cone varieties,homogeneous ideals, the cone over a variety. | VIII.1-4. |
| Wed. 5/1 | Projective geometry. | VIII.1-4. |