# 18.725: Introduction to Algebraic Geometry

Update: click here for a much later version (really, a distant descendant)

The description in the course guide: "Introduces the basic notions and techniques of modern algebraic geometry. Algebraic sets, Hilbert's Nullstellensatz and varieties over algebraically closed fields. We relate varieties over the complex numbers to complex analytic manifolds. For varieties of dimension one (i.e. curves) we discuss the genus, divisors, linear series, line bundles and the Riemann-Roch theorem." Johan de Jong will be teaching the follow-up course in the spring. The class is in 24-407.

Here are various notes in dvi, ps, and pdf formats. (If you would like tar archives, just let me know; due to limited diskspace, I had to remove them from this page.) I've added all the corrections I've found (or been told about) on Dec. 30, 1999, so they are in as final form as they'll ever be.
• On the first day (Sept. 9), I gave out two handouts, one with information about the course (dvi, ps, or pdf), and one with fun problems in algebraic geometry to pique your interest (dvi, ps, or pdf).
Here's a rather detailed summary of the first lecture (dvi, ps, or pdf).
• On the second day (Sept. 14), I gave out the first problem set (dvi, ps, or pdf), due in class on Sept. 21.
Here's a detailed summary of the second lecture (dvi, ps, or pdf).
• Here's a detailed summary of the third lecture (dvi, ps, or pdf).
• Here's a detailed summary of the fourth lecture (dvi, ps, or pdf). I'm planning on revising it slightly in light of some good questions in class, and reposting it here. But there won't be any really substantive changes.
Here's the second problem set (dvi, ps, or pdf), due in class on Sept. 28.
• Here's a detailed summary of the fifth lecture (dvi, ps, or pdf) (Sept. 23).
• Here's a detailed summary of the sixth lecture (dvi, ps, or pdf) (Sept. 28).
Here's the third problem set (dvi, ps, or pdf), including minor corrections. By popular demand, it will be due in class on Thursday Oct. 7 (not Tuesday Oct. 5 as it says on the earlier version of the set).
• Here's a detailed summary of the seventh lecture (dvi, ps, or pdf) (Sept. 30).
• Here's a detailed summary of the eighth lecture (dvi, ps, or pdf) (Oct. 5). Note corrections to minor errors in ps3, that I mentioned in my e-mail of Oct. 5.
• Here's the ninth lecture (dvi, ps, or pdf) (Oct. 7).
Here's the fourth problem set (dvi, ps, or pdf), due Oct. 14. Don't do the scheme problems on the version handed out in class; they will reappear next week (after I introduce schemes).
• Here's the tenth lecture (dvi, ps, or pdf) (Oct. 12). It was an optional lightning-fast introduction to schemes. We'll return to prevarieties on Thursday.
• Here's the eleventh lecture (dvi, ps, or pdf) (Oct. 14).
Here's the fifth problem set (dvi, ps, or pdf), due Oct. 21.
• Here's the twelfth lecture (dvi, ps, or pdf) (Oct. 19).
• Here's the thirteenth lecture (dvi, ps, or pdf) (Oct. 21).
Here's the sixth problem set (dvi, ps, or pdf), due Oct. 28.
• Here's the fourteenth lecture (dvi, ps, or pdf) (Oct. 26).
• Here's the fifteenth lecture (dvi, ps, or pdf) (Oct. 28).
Here's the seventh problem set (dvi, ps, or pdf), minus 2 figures (which are in Hartshorne p. 36), due Nov. 4.
• Here's the sixteenth lecture (dvi, ps, or pdf) (Nov. 2).
• Here's the seventeenth lecture (dvi, ps, or pdf) (Nov. 4).
Here's the eighth problem set (dvi, ps, or pdf), due Nov. 11 (at noon, at my office 2-271).
• Here's the eighteenth lecture (dvi, ps, or pdf) (Nov. 9).
Here's the ninth problem set (dvi, ps, or pdf), due at some point in the indefinite future.
• No class on November 11.
• Here's the nineteenth lecture (dvi, ps, or pdf) (Nov. 16).
• Here's the twentieth lecture (dvi, ps, or pdf) (Nov. 18).
Here's the tenth problem set (dvi, ps, or pdf), due at some point in the indefinite future (probably Tues. Nov. 30).
• Here's the twenty-first lecture (dvi, ps, or pdf) (Nov. 23).
• Here's the twenty-second lecture (dvi, ps, or pdf) (Nov. 30). Owen Jones from Imperial College London has caught these typos. (Thanks Owen!) On the second page in the paragraph beginning "let's consider this as a rational" I have "g_0 (z_0) = f_{01} f_1 (y_0)" but the f_1 should be a g_1. In the same paragraph at the end of that line of working, "(z_0 - 1)" should be "(1 - z_0)". On page 4 of the same document in the paragraph starting "Now we'll deal" we have P^1 injecting into P^m, when we actually mean P^n.
• Here's the twenty-third lecture (dvi, ps, or pdf) (Dec. 2). Owen points out: Just under the definition of the Picard group I've forgotten the inverses when defining the mth tensor power of an invertible sheaf for m negative.
Here's the eleventh problem set (dvi, ps, or pdf), due Thurs. Dec. 9. (Thanks to a good question in Thursday's class, I've added a small part to problem 4.)
• Here's the twenty-fourth lecture (dvi, ps, or pdf) (Dec. 7).
Here's the twelfth problem set (dvi, ps, or pdf), due Monday Dec. 13 (in my office at noon). (In the version handed out, I omitted two dr's from problem 2.)
• Here's the twenty-fifth (and final) lecture (dvi, ps, or pdf) (Dec. 9). (Missing table of contents.)
• Here's a proof of Riemann-Roch and Serre duality (for curves) that I gave in the Baby Algebraic Geometry Seminar (dvi, ps, or pdf) (Feb. 11), that fits well at the end of these notes. (Later this week, after I've had a chance to make corrections, I'll put all of these notes in a tarred file for ease of downloading.)
• WARNING: These notes are not intended to be absolutely complete; they're a cleaned-up version of my notes to myself for each class. You will often need to supplement them by looking in references, or asking me questions. Also, I hope to correct errors in the notes every so often, but only when I have time (which isn't often). I'm hoping that access to these notes (errors and all) is better than access to no notes.
• Some fun questions that came up in discussions with Anders Buch (who is teaching this course in fall 2000). Is every affine open subscheme of an affine scheme a distinguished open? Given a morphism from an affine scheme to another scheme, must the image necessarily lie in an affine open of the target? (For answers, just ask.)