Open Questions: Algebraic Geometry

See also: Number theory -- Geometry and topology

Math Forum Internet Mathematics Library: Algebraic Geometry: Alphabetized list of links with extensive annotations.
Open Directory Project: Algebraic Geometry: Categorized and annotated algebraic geometry links. A version of this list is at Google, with entries sorted in "page rank" order.
Algebraic Geometry: A very good list of links, by Tyler J. Jarvis.
Galaxy: Algebraic Geometry: Categorized site directory. Entries usually include descriptive annotations.

Algebraic geometry: Article from Wikipedia.
Rational and integral points on higher dimensional varieties: An outline (literally) of some of the conjectures and open problems in this area of arithmetic algebraic geometry. Part of the Workshop Website Network of the American Institute of Matheamtics. Includes a detailed glossary.
A sheaf-theoretic refinement of the Tate conjecture: Preprint of a technical paper by Bruno Kahn. DVI and PS formats available.
Algebraic Geometry: Course notes from an introductory course, available in DVI, PS, and PDF formats. By J. S. Milne.
Lectures on Etale Cohomology: Course notes from an introductory overview, available in DVI, PS, and PDF formats. By J. S. Milne.
Abelian Varieties: Course notes from an introduction to the geometry and the arithmetic of abelian varieties available in DVI, PS, and PDF formats. By J. S. Milne.

The Hodge Conjecture: Brief description of the problem at the Clay Mathematics Institute site by Pierre Deligne. (A more complete description is available as a PDF file.)
Hodge conjecture: Article from Wikipedia.

Arithmetic on Curves
Barry Mazur
Bulletin of the AMS, April 1986, pp. 207-259: Diophantine problems in number theory can usually be expressed in terms of, or related to, arithmetic questions about algebraic curves. Faltings' proof of Mordell's conjecture is a key result, which deals with important concepts of algebraic geometry, such as abelian varieties and jacobians.
The Proof of the Mordell Conjecture
Spencer Bloch
The Mathematical Intelligencer, Vol. 6, No 2, 1984, pp. 41-47: Gerd Faltings's 1983 proof of the Mordell conjecture is one of the most important accomplishments in the field of artithmetic algebraic geometry. The proof also settled, as essential steps, several other important questions as well, including conjectures of John Tate and I. R. Shafarevich.

Karen E. Smith; Lauri Kahanpää; Pekka Kekäläinen; William Traves – An Invitation to Algebraic Geometry
Springer-Verlag, 2000: Modern algebraic geometry is a notoriously difficult subject. It's founded upon (and was often the motivation for) some of the most abstract concepts of abstract algebra. But the authors do a fine job, in a short book, of describing the basic principles of the subject, some of the important developments of the last 100 years, and some of the most interesting open problems. And the prerequisites are few – just a basic course in linear algebra.