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 RiemannRoch theorem." Johan de Jong will be
teaching the followup course in the spring. The class is in
24407.
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 email 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 lightningfast 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 2271).
 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 twentyfirst lecture
(dvi, ps,
or pdf)
(Nov. 23).
 Here's the twentysecond 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 twentythird 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 twentyfourth 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 twentyfifth (and final) lecture
(dvi, ps,
or pdf)
(Dec. 9). (Missing table of contents.)
 Here's a proof of RiemannRoch 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 cleanedup 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.)
Here are other links:

The notes to Olivier Debarre's introductory course in algebraic
geometry are available from
his homepage (in french).

The notes to Igor Dolgachev's
introductory course in algebraic geometry are available from
his
lecture notes page.
 Bernd Sturmfels and Greg Smith developed some great computational
problems to accompany an introductory course. They are available
here.
Back to my home page.
Ravi Vakil
Department of Mathematics Rm. 2271
Massachusetts Institute of Technology
77 Massachusetts Ave.
Cambridge MA USA 02139
Phone: 6172532683 (but email is better)
Fax: 6172534358
Email: vakil@math.mit.edu