Math 631: Introduction to Algebraic Geometry

Professor Karen E. Smith
East Hall 3074

FALL 2012
MWF at 1 pm in 4096 East Hall.
Office hours immediately after class on mondays and 11 am on Thursdays; also by appointment.

Math 631 is a first introduction to classical algebraic geometry over the complex numbers for "beta level" PhD students in math. Beta level means the student is expected to have mastered the material of the alpha courses at a fairly sophisticated level (Math 295-396 and 512-513 is definitely *not* sufficient preparation).

Prerequisites: Math 591, Math 593, Math 594, Math 596, and Math 614. Math 537 (or some working knowledge of manifolds, tangent spaces, etc) is also recommended. Math 614 can be taken simultaneously if the student is strong otherwise in algebra, willing to struggle through some ideas on their own, and is comfortable taking the proofs of many statements on faith. (See below for more on prerequisites.)

The Daily Update, a summary of what was discussed in class each day, including assignments and quiz announcements. Lecture Notes, written up by student Daniel Hast.

Quizzes: Quiz 1 , Quiz 2 , Quiz 3, in advance! , Quiz 4 , Quiz 5 , Practice Quiz 6 , Quiz 7 ,

Problem Sets: Set 1 , Set 2 , Set 3 , Set 4 , Set 5 , Set 6 , Set 7 , Set 8 , Set 9 , Set 10 , Set 11 (Note: This last set is basically 1.5 times as long as the others, as I combined the last two sets into one. On the other hand, you have 3 weeks to do it.).

Syllabus for Math 631: affine and projective varieties, Hilbert's Nullstellensatz, the Zariski topology, the sheaf of regular functions, regular and rational maps, dimension, the Zariski tangent space, the concept of smoothness, degree, the Hilbert polynomial, blowing up, divisors, line bundles and maps to projective space, the Riemann-Roch formula for curves. Considerable attention will be paid to the rich examples of classical algebraic geometry: Grassmannians, flag varieties, curves, Segre and Veronese maps, blow-ups, quadrics, determinantal varieties. We will also touch on the idea of schemes and of varieties as functors, though these concepts will be developed only as examples, not formally or precisely.

Expectations: This will be a ``seriously graded" course, moving at a fast pace with a very high workload even for the best prepared students. There will be a long problem set every week, as well as an (usually unannounced) weekly quiz and a final paper. Attendance is required, as is participation in class discussion. Students who are already PhD candidates can negotiate with me a different set of expectations.

About homework: Ideally, these will be texed-up. Students may work together provided collaboration is acknowledged. Students may also turn in "joint" problem sets, but must follow the guidelines discussed in class. Use of websites such as MathStackExchange for "hints" on homework is considered CHEATING and will be promptly and vigorously discplined. Students who are aware of others cheating are *expected* to report this behavior.

Text: Shafarevich, Basic algebraic geometry (Vol I)", by Igor R. Shafarevich

More recommended Reading:

  • Retired UM professor Jim Milne's 631 notes (pretty comprehensive with lots of background material, fairly close to my style).
  • Joe Harris's "Algebraic Geometry, a first course" (tons of great examples and intuition).
  • Karen Smith's Invitation to Algebraic Geometry (a very gentle discussion of the main ideas).
  • Hochster's notes for Math 614
  • David Eisenbud's ``Commutative algebra with a view toward Algebraic Geometry."
  • Hartshorne's "Algebraic Geometry" (the standard reference book).
    I do not think Hartshorne's book gives a deep enough introduction to classical algebraic geometry, but it does have the needed material for Math 632 in the winter (schemes and cohomology) and is a standard textbook, so you might want to invest in it at some point if you plan to continue. [By the way, more inportant prerequisites for 632 include 614 (commutative algebra) and 592 (or some other homological algebra/ algebraic topology course).]

    More on prerequisites: Mere "exposure" to the prerequisite topics is not sufficient. I expect students to have a good working knowledge of advanced linear algebra: for example, you can easily determine whether two quadratic forms over the complex numbers are the same up to change of coordinates, you can work fluently with tensor and exterior products (if not, take Math 593 instead). I expect students to be comfortable with point set topology and basic smooth manifolds: for example, you can get started right away if asked to prove a given topological space is compact, given a description of its open sets, and you could compute equations for the tangent space to a manifold embedded in R^n, say, if the manifold was defined in simple as in a good multi-variable calculus class (if not, take Math 591 or 437 or 537). You should have have some intution for complex analytic functions: for example, you can immediately describe all the analytic functions on a closed disc with infinitely many zeros, and you know in what sense is a complex analytic function "the same" as a convergent power series? If not, take Math 596). You should also have a good feeling for groups acting on different sets: for example, confronted with a group acting on some polynomial ring or some topological space, you could get started on looking for some invariant polynomials or some orbits, and you can easily identify the galois group of many common field extensions (if not, take Math 594 or Math 591). Ideally, students will also be comfortable with a substantial amount of commutative algebra: for example, you immediately know why a finitely generated algebra over field must be isomorphic to a polynomial ring modulo an ideal, you know about UFDs and PIDs, and you've studied localization and integral extensions (if not, take Math 593,594, or 614).

    I should add that the lectures will assume somewhat less sophistication than the homework; so you may be able to follow the lectures (at least for a while) without being able to do many problems. If you think you "need an A" please do not ignore the stated prerequisites.

    The problem sets are the main point of the course. Your performance on the homework assignments essentially determines your grade, as long as it is confirmed by similar performance in class (quizzes) and the final paper. In various experiments done over the years by myself and my colleagues, the statement "student performance on the homework assignments essentially determines their grade" is true no matter how much the homework is formally weighted. So just do the homework! There is no point in cheating, for example, even mindlessly copying others solutions. But in terms of specific numbers, let's say for a base-line first approximation at a grade: homework counts 40%, quizzes 40% and final paper 20%. But of course, an amazing final papers, perfect quizzes, deep curiosity and work ethic as displayed in office hours, even deep commitment to the material (for example by always redoing quizzes) or any other mitigating factor, will be taken into account. You are a "whole student," not a number, so I prefer to grade somewhat holistically. There is no curve for the grades. I use a scale of B- to A+ for students who actually come to class in turn in work, but (assuming this class is typical) most students wind up doing A work, so I expect that probably more than half the students will get some kind of A (usually A- or A, mostly, with a few A+'s).