Math 797W: Algebraic geometry
Fall 2016
Instructor: Paul Hacking, LGRT 1235H, hacking@math.umass.edu
Meetings
Classes: Tuesdays and Thursdays, 8:30AM--9:45PM in LGRT 1114.
Office hours: Tuesdays 3:00PM--4:00PM and Wednesdays 2:30PM--3:30PM in my office LGRT 1235H.
Prerequisites
Commutative algebra (rings and modules) as covered in 611-612. Some prior experience of manifolds would be useful (but not essential).
Homework
HW1 pdf. Due Thursday 9/29/16.
HW2 pdf. Due Tuesday 10/25/16.
HW3 pdf. Due Thursday 12/1/16.
HW4 pdf. (Will not be graded).
Overview of course
Algebraic geometry is the study of geometric spaces locally defined by polynomial equations. It is a central subject in mathematics with strong connections to differential geometry, number theory, and representation theory. This course will be a fast-paced introduction to the subject with a strong emphasis on examples. In the algebraic approach to the subject, local data is studied via the commutative algebra of quotients of polynomial rings in several variables. Passing from local to global data is delicate (as in complex analysis) and is either accomplished by working in projective space (corresponding to a graded polynomial ring) or by using sheaves and their cohomology. Topics will include projective varieties, singularities, differential forms, line bundles, and sheaf cohomology. Examples will include projective space, the Grassmannian, blow-ups and resolutions of singularities, algebraic curves of low genus, and surfaces in projective 3-space.
Useful references
Introductory texts
Undergraduate algebraic geometry, M. Reid, googlebooks.
Elementary introduction to algebraic geometry. Very accessible.
Basic algebraic geometry 1, I. Shafarevich, googlebooks. Fairly extensive introduction with few prerequisites.
The red book of varieties and schemes, D. Mumford, googlebooks. Classic text.
Algebraic geometry I. Complex projective varieties, D. Mumford, googlebooks. An introduction to classical algebraic geometry using a combination of algebraic, analytic, and topological methods
Algebraic geometry: a first course, J. Harris, googlebooks. Recent book with lots of examples.
More advanced texts
Algebraic geometry II, D. Mumford and T. Oda, pdf ("penultimate draft"), googlebooks. Expanded version of the red book (see above) including sheaf cohomology.
Algebraic geometry, R. Hartshorne, googlebooks. Standard text covering modern techniques in algebraic geometry. Rather intimidating for the beginner.
Principles of algebraic geometry, P. Griffiths and J. Harris, googlebooks. Describes the analytic approach to algebraic geometry. Full of instructive examples.
Hodge theory and complex algebraic geometry I, C. Voisin, googlebooks. Thorough development of the analytic approach (more careful than Griffiths-Harris, but fewer examples).
Commutative algebra
Undergraduate commutative algebra, M. Reid, googlebooks. An introductory text.
Introduction to commutative algebra, M. Atiyah and I. MacDonald, googlebooks. Classic text (very concise).
Commutative ring theory, H. Matsumura, googlebooks. Useful reference.
Commutative algebra with a view toward algebraic geometry, D. Eisenbud, googlebooks. Comprehensive text including the geometric point of view.
This page is maintained by Paul Hacking hacking@math.umass.edu