18.726: Algebraic Geometry
This is the home page for the MIT class
Geometry) for the spring 2005 semester. The class meets Tuesdays and
Thursdays 11-12:30 in 4-261.
The class is the second
in a two-semester graduate-level sequence in algebraic geometry
beginning with 18.725.
sequence is intended for students planning research either in algebraic
or in a neighboring area in which algebraic geometry plays a major role
(algebraic topology, representation theory, arithmetic geometry, etc.).
The topics to be covered in 18.726 include schemes, properties of
morphisms of schemes, coherent sheaves, and sheaf cohomology; for more
details, see the syllabus below.
- Lecture calendar (and note files)
- New: Virtual office hours (answers to
- Morphisms of schemes crib sheet (TeX,
- Problem Set 1 (due February 10). Changes
made after posting: I'm dropping the restriction that you do only two
problems from II.1, since I didn't get as far in class as I intended to
on Thursday. (You're still required to do the problems marked
"Required".) Also, in problem 6(c), you may interpret "describe" to
mean "describe in words"; don't try to exhibit explicit generators for
the corresponding ideals.
- Problem Set 2 (due February 17).
- Problem Set 3 (due February 24). Don't forget that
there is no class on the 22nd, since MIT runs on a Monday schedule that day.
Changes made after posting: remove Required status from any problems mentioning
properness, since that wasn't covered in class on Thursday.
- Problem Set 4 (due March 3).
- Problem Set 5 (due March 10).
- Problem Set 6 (due March 17).
- Problem Set 7 (due March 31). Corrections: problem 1
is a repeat, so don't do it again if you did it before. In problem 6, assume the characteristic of the field is zero. (Remind me to give you the sheaf
cohomology version of this later; that works in characteristic not 2.) Also,
book typo: in II.8.5(a), the map Z -> Pic X should map to Pic X~ (the blowup).
- No problem set 8; instead, see
the suggested problems in homological algebra.
- Problem Set 9 (due April 14). Yes, I'm going to
keep numbering this way even though there isn't really a PS8.
- Notes on the relation between Cech and sheaf
cohomology (from April 14 lecture) Corrections: on page 1, "multiplication
by f" should be "dot product by f"; last line of page 1, "Hom_A(K^.(f),M)"
should be "Hom_A(K_.(f),M)".
- Problem Set 10 (due April 28; we skip a week
because of Patriots Day).