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 this page.

Here is a more detailed course info sheet.

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, 5.2(a).

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 spring break.

Homework 6 due Friday 4/10 is Harthshorne: II.7.3, 7.7(b,c), 7.12 + the following problem: 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 term.