This course provides an introduction to algebraic geometry. We will begin by studying affine varieties, projective spaces, projective varieties and maps between them. We will establish a dictionary between geometry and algebra; one of the important tools here will be Hilbert's Nullstellensatz. Next, we will look at plane curves. We will prove Bezout's theorem and show some of its applications. We will define the addition law on plane cubic curves - a more detailed discussion of elliptic curves will be given. The end of the course will present more advanced topics, such as the existence of 27 lines on the cubic surface.

**Instructor:** Dragos Oprea, oprea "at" math.you-know-where.edu,
Room 382D (2nd floor).

**Lectures:** MWF,
11am-11:50am.

**Office hours: ** Th 2:15-4:15, 382D

**Course
Assistant:**Yu-jong Tzeng, yjt "at" math.you-know-where.edu, Room
380L

**Office
hours:** M 3-5pm, TTh 10:30-12, W 4:10-5:10.

**
Textbook:** Miles
Reid - Undergraduate algebraic geometry.

** Prerequisites:** This
course is intended for 3rd or 4th year undegraduate students. I expect
that people taking the class will have some background in algebra at the
level of Math 120. For instance, I will assume familiarity with fields,
rings, polynomial rings etc. It will be useful to know some complex
analysis at the level of 116. Familiarity with basic point-set topology
may be beneficial but not required. Some knowledge of manifolds will be
even better but definitely not required.

** Exams:** There will be a take-home midterm and a
take-home final.

** Problem Sets:** There will be weekly problem sets, usually due on
Friday in class. The problem sets will be posted online. Group work is
encouraged, but you have to hand in your own write up of the homework
problems.

** Final Grades:**

- Problem sets - 30 percent
- Take-home midterm 30 percent.
- Take-home final Exam - 40 percent

** Important dates:**

- Drop deadline: April 27.
- Withdrawal deadline: May 26.
- Last day of classes: June 4.

** Lecture Summaries**

**Lecture 1:**Introduction to algebraic geometry. Connections between algebraic geometry and other fields of math. Affine algebraic sets. Affine plane curves.-
**Lecture 2:**Affine algebraic sets (continued). Ideals in polynomial rings. Noetherian rings. Examples of Noetherian rings. -
**Lecture 3:**Hilbert basis theorem. Topological spaces. Zariski topology on A^n. **Lecture 4:**The ideal of an algebraic set. Weak and strong Nullstellensatz. Radical ideals. Examples.**Lecture 5:**I started the proof of the weak Nullstellensatz. I defined the resultant of two polynomials as the determinant of the Sylvester matrix. I showed that if two polynomials have a common factor then their resultant is zero. I showed that the resultant is a combination of the two polynomials.-
**Lecture 6:**More about resultants. R_{f,g} is the product of differences of the roots of f and g. Multiplicativity of resultants e.g. R_{f_1f_2, g}=R_{f_1, g} R_{f_2, g}. If A is UFD, R_f,g=0 iff f and g have a common factor. For polynomials in n variables, we can define the resultant in n different ways for polynomials in n variables. These are in general different polynomials. Resultants of homogeneous polynomials are homogeneous of degree deg(f) deg(g). **Lecture 7:**The proof of the weak Nullstellensatz. The strong Nullstellensatz.**Lecture 8:**Irreducible affine algebraic sets. Prime ideals. Maximal ideals. Examples.**Lecture 9:**Irreducible components. Examples of decompositions into irreducible components.**Lecture 10:**Dimension. Morphisms and isomorphisms between affine algebraic sets. Coordinate rings. Morphisms/isomorphisms induce homomorphisms/isomorphisms between coordinate rings.**Lecture 11:**I proved that morphisms between coordinate rings come from morphisms of affine algebraic sets. I defined rational functions. I defined regular points of rational functions. I defined rational maps.**Lecture 12:**Quasi-affine varieties. Rational maps and morphisms between quasi-affine varieties.**Lecture 13:**Projective space. Projective nullstellensatz. Lines in projectives space.**Lecture 14:**Conics in projective space. Showed that there is only one conic up to change of coordinates. Discussed how the two conics in A^2 "close up" to only one conic in P^2. I defined rational maps, morphisms, isomorphisms etc for projective varieties.**Lecture 15:**Examples of morphisms. There are no constant regular maps on P^1.**Lecture 16:**Homogeneous coordinate rings of isomorphic projective varieties need not be isomorphic. Veronese morphisms. Segre morphisms. Quadrics in P^3 admit two rulings.**Lecture 17:**Segre embedding. P^n times P^m is a projective variety cut out by quadrics. Subvarieties of P^n times P^m are given by bihomogeneous equations. Stated main theorem of projective varieties. Used it to derive that projective varieties have no regular functions defined anywhere. Reduced the proof of the theorem to the projection from P^n\times P^m\ to P^m.**Lecture 18:**Proved that image of projective sets under morphisms are projective.**Lecture 19:**Singularities of affine and projective curves. Multiplicities. Ordinary singularities. Examples.**Lecture 20:**Blowups. Example: nodes are resolved by blowups, tacnodes become nodes under blowup. Inflection points. I showed that any curve of degree 3 has at least an inflection point. I showed that the point at infinity is an inflection point for the elliptic curve E_lambda.**Lecture 21:**Every non-singular cubic curve is isomorphic to an elliptic curve E_lambda. However, some of the E_lambda's may be isomorphic to each other. The j-invariant.**Lecture 22:**The j invariant of the Fermat cubic. The elliptic curve which solves the congruent number problem. Outline of the addition law on the elliptic curve.**Lecture 23:**Weak Bezout theorem. A line and a cubic intersect in 3 points counted with multiplicity. The group law on the cubic curve. Examples.**Lecture 24:**Stated the main lemma: 2 cubics intersecting in 9 points, if a third cubic passes thoug 8 points of intersection it must pass though the 9th as well. Deduced associativity on the elliptic curve. Pascal's theorem.**Lecture 25:**Proved the main lemma. Elliptic curve cryptography.**Lecture 26:**Overview of rational points on elliptic curves: Mordell-Weil, Faltings, Mazur theorems. BSD conjecture. Modern algebraic geometry.

- Notes on Hilbert's Nullstellensatz PDF.

- Notes on irreducibility and dimension PDF. Updated 04/21/08.

- Complete course notes courtesy of David Philipson PDF.

** Homework 1** due Monday, April 14, in class PDF.

**Solutions:**PDF

** Homework 2** due Friday, April 18, in class PDF.

**Solutions:**PDF

** Homework 3** due Friday, April 25, in class PDF.

**Solutions:**PDF

** Homework 4** due Friday, May 2, in class PDF.

**Solutions:**PDF

** Midterm ** due Wednesday, May 7, in class PDF.

**Solutions:**PDF

** Homework 5** due Friday, May 16, in class PDF.

**Solutions:**PDF

** Homework 6** due Friday, May 23, in class PDF.

**Solutions:**PDF

** Homework 7** due Friday, May 30, in class PDF.

**Solutions:**PDF

** Final Exam** due Friday, June 6 (or the latest on Monday June 9
at 11:30AM), in my mailbox PDF.