This is the page for the course 18.726, Algebraic Geometry.
The course meets MW 1:30-3 at 36-112.
We will mostly follow Hartshorne's
textbook with some additions and variations, homeworks will be posted at
Here is a more detailed course info
The first assignment due Friday 2/13 is from Harthshorne, chapter II: 1.10, 2.6, 2.8, 2.17(b), 2.19.
In problem 1.10, in addition to answering the question in the book
produce an example of a system of sheaves for which the direct limit in the
category of presheaves is not a sheaf.
Homework 2 is due Friday 2/20 at 3pm.
Homework 3 due Friday 2/27 is from Harthshorne: II.5.2, 5.8; III.3.1, 3.2, 4.7.
Homework 4 due Friday 3/6 is Harthshorne: II.4.6; III.3.8, 4.8(c.e), 4.9,
Homework 5 due Friday 3/13 is Harthshorne: III.6.2, 6.5, 7.1, 8.3 +
the following problem:
Let X be the affine line with one double point (Example II.2.3.6).
Describe the Picard group of X and for each line bundle on X verify
from definitions that the line bundle is not ample. Moreover, for each such line
bundle find a sheaf
F on X such that any twist of F by a positive power of the line bundle
is not generated by global sections.
MODIFICATION to homework 5: III.6.2 and III.8.3 can be handed in after the
Homework 6 due Friday 4/10 is Harthshorne: II.7.3, 7.7(b,c), 7.12 + the
Let X be a cuspidal cubic in the projective plane over complex numbers;
more precisely, let X be given by the equation y^2z=x^3. Let Y be the
product of X by
the affine line A^1. Give an example of a line bundle on Y whose
restrictions to different fibers of the second projection are
non-isomorphic. Conclude that the natural map from Picard group of X to
the Picard group of the product of X by the affine line may not be an
isomorphism when X is singular.
Homework 7 is due Friday 4/21.
Homework 8 is due by the end of the