Elementary Algebraic Geometry.
Classes: 3107 Etcheverry, Tue Th 8-9:30. Office hours: Tue 3:40-5, Fri 2-3 (813 Evans).
Main textbooks:
[UAG] Miles Reid, Undergraduate Algebraic Geometry [pdf] and
[UCA] Miles Reid, Undergraduate Commutative Algebra (not freely available online).
Additional texts: [Ga] Andreas Gathmann, Algebraic Geometry [pdf]; [Smi] Justin R Smith, Introduction to Algebraic Geometry [pdf]; [Va] Ravi Vakil, Algebraic Geometry [pdf].
Practice Final , Practice Midterm.
We will cover the first halves of [UAG] and [UCA], possibly, with some additional material depending on how the class goes. Tentative topics to be covered: rings, ideals and modules; Noetherian rings; Spec of a ring; Nullstellensatz; Bezout's theorem; complex curves; affine and projective varieties, their morphisms. Possible extra topics: 27 lines on the cubic surface, introduction to toric varieties.
HW 40%, midterm 30%, final 30%. Each of the three scores is curved separately. The score for the final replaces the score for the midterm if higher. One worst HW score will be dropped.
HW is due on paper by 10:30am of the due date. Submission options: you can bring it to me in class personally, or ask someone else to handle it to me in class, or bring it to my office (813 Evans) between 9:30 and 10:30.
You are welcome to typeset your HW on a computer. If you do so, please print it out and submit as explained above, and do not send it to me by email. In exceptional circumstances (for example, if you are out of town), I may agree to accept HW by email; please seek advance permission.
I may randomly choose a subset of the assigned problems that will be checked, and quickly look through the others. You will not know beforehand which problems I'm going to check.
You are welcome collaborate on homework problems, but your written solutions must be your own work. This means, in particular, that you cannot consult anyone else's solutions when writing your work.
| Date | Summary | Literature | Hw | Remarks |
| 8/23 | Affine varieties. x^2+y^2=1 over Q, R and C. Affine conics. | UAG 1.1-1.3 | ||
| 8/28 | Example: twisted cubic. Projective space, projective transformations. PGL(n+1) is (n+2)-transitive. | Ga Ex 0.1.9, Va Lec 3, 4 | HW 1 | |
| 8/30 | Varieties: passing from affine to projctive charts and back. Classification of projective conics. Bezout theorem, statement. Application: counting the number of intersections in an affine space. Intersection patterns of 2 conics. | Va Lec 5, UAG 1.5, 1.6, 1.8-1.12, example not from the literature on the last page of my notes | ||
| 9/4 | Rational parametrization of a conic. Proof of Bezout when one curve is a conic. Sketch proof of general Bezout for curves. Gradient. Tangent space to a hypersurface, smoothness. | UAG 1.9, Smi 1.3 and beginning of 1.4, Va Lec 6 | HW 2 | |
| 9/6 | Euler's formula. Solving equations mod p^k iteratively. Rings, ideals, quotients. Coordinate ring of an affine variety (definition). Examples: quotients of k[x,y] by (x,y), (x), (xy). | Va Lec 6, UCA 1.1 and Exercise 1.3 | ||
| 9/11 | Prime and maximal ideals. Spec. Examples: Spec of k[x,y] and Z[x]. | UCA 1.2-1.5 | HW 3 | |
| 9/13 | Zorn's lemma, existence theorems for maximal and prime ideals, nilradical, radical, local rings. | UCA 1.7-1.13 | ||
| 9/18 | Examples of local rings. Localization at a prime ideal. Tangent space of an affine variety; equivalent definition as m/m^2. Smoothness. Jacobi matrix. | UCA 1.13-1.15, UAG 6.8, Ga 4.4 | HW 4 | |
| 9/20 | k[x_1...x_n]/(f_1,...,f_k) is local if {f_i=0} is a point (without proof). Tangent spaces m/m^2 are preserved under localization. Modules. Generators. | Smi Thm 3.3.7 (p. 123), UCA 2.1-2.5 | ||
| 9/25 | Cayley-Hamilton theorem, determinant trick, Nakayama's lemma. | UCA 2.6-2.8 | HW 5 | |
| 9/27 | Short exact sequences, split condition. Noetherian rings. | UCA 2.9-3.1 | ||
| 10/2 | Comparison: rings of smooth or analytic functions are non-Noetherian. Noetherian modules and exact sequences. Hilbert's basis theorem. | UCA 3.2-3.6 | HW 6 | |
| 10/4 | Reminder: basics of ring extensions. Algebra over a ring. Integral element of an algebra over a ring. Examples. | UCA 4.1 | ||
| 10/9 | Integral elements, tower laws, integral closure, normalization. Facts about normalization for curves (without proof). | UCA 4.2-4.5 | HW 7 | |
| 10/11 | Noether normalization. Weak Nullstellensatz about field extensions. | UCA 4.6-4.9 | ||
| 10/16 | MIDTERM | |||
| 10/18 | Nullstellensatz. | UCA 5.1-5.6 | ||
| 10/23 | Corollaries of Nullstellensatz on subvaries. Zariski topology. Decomposition into irreducible subvarieties. | UCA 5.7-5.11 | HW 8 | |
| 10/25 | Zariski topology on the Spec of a general ring. Localization for a general ring. | UCA 5.11-5.13, 6.1-6.2 | ||
| 10/30 | Localizing curves at points: examples. Modules of fractions. Localization is exact. Localization commutes with taking quotients. | UCA 6.4-6.7 | HW 9 | |
| 11/1 | Artinian rings, structure theorem. | Brad Drew's AG notes Lec 12, up to Prop 7 | ||
| 11/6 | Coprime ideals, Chinese remainder theorem for rings, Artinian rings as finite-dimensional algebras. | Prop 1.10 from Atiyah-McDonald, Brad Drew's AG notes Lec 12, Propositions 7-9. | HW 10 | |
| 11/8 | Artinian rings are Noetherian. A fin. gen. algebra is Artinian iff it is finite-dimensional. k[x_1...x_n]/I is Artinian if V(I) consists of finitely many points. | See eg Prop 2.3 from here for Artinian implies Noetherian, and Brad Drew's AG notes Lec 12, Prop 10-11 for the rest. | ||
| 11/13 | Intersection multiplicity, examples. Affine version of Bezout thm, with a proof. | See my notes for most of the lecture, and Gathmann's Alg Curves Chapter 2 up to Lemma 2.5 | ||
| 11/15 | Intersection multiplicity and short exact sequences. Algorithm for computing intersection multiplicities. Examples. | Gathmann's Alg Curves Chapter 2 Example 2.6-Rmk 2.15 | ||
| 11/20 | Cancelled due to air conditions | HW 11 | ||
| 11/22 | Thanksgiving holiday | |||
| 11/27 | Graded rings and homogeneous ideals. Hilbert function of a homogeneous ideal, examples. Theorem-definition about the Hilbert polynomial, statement. | Gathmann's Alg Geom Chapter 6 Dfn 6.6 - Lemma 6.10 and Alg Geom Chapter 12 beginning - Prop/Defn 2.11 | ||
| 11/29 | Theorem about the Hilbert function and Hilbert polynomial. Bezout's theorem revisited. | The rest of Gathmann's Alg Geom Chapter 12. |