18.726: Algebraic Geometry

The class is the second in a two-semester graduate-level sequence in algebraic geometry beginning with 18.725. This sequence is intended for students planning research either in algebraic geometry 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.

• Syllabus
• Lecture calendar (and note files)
• New: Virtual office hours (answers to key questions)
• Morphisms of schemes crib sheet (TeX, pdf, dvi, ps)
• 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).