University of Illinois at Urbana-Champaign
College of Liberal Arts and Sciences
Department of Mathematics

Mathematics 422, section B1
Algebraic Geometry
9-9:50 AM MWF
343 Altgeld Hall

Professor Daniel R. Grayson
Office: 357 Altgeld Hall
Phone: 3-6209


Algebraic geometry is the geometric study of solutions of systems of polynomial equations in several variables. It plays a central role in much of modern mathematics, and an understanding of its basic concepts is increasingly important.

The textbook:

  • The geometry of schemes, by David Eisenbud and Joseph Harris, 2000, Springer, Graduate Texts in Mathematics, 197.
    From the introduction: No one can doubt the success and potency of the scheme-theoretic methods. Unfortunately, the average mathematician, and indeed many a beginner in algebraic geometry, would consider our title, "The Geometry of Schemes", an oxymoron akin to "civil war". The theory of schemes is widely regarded as a horribly abstract algebraic tool that hides the appeal of geometry to promote an overwhelming and often unnecessary generality. By contrast, experts know that schemes make things simpler. The ideas behind the theory - often not told to the beginner - are directly related to those from the other great geometric theories, such as differential geometry, algebraic topology, and complex analysis. Understood from this perspective, the basic definitions of scheme theory appear as natural and necessary ways of dealing with a range of ordinary geometric phenomena, and the constructions in the theory take on an intuitive gometric content which makes them much easier to learn and work with. It is the goal of this book to share this "secret" geometry of schemes.
  • Additional references:

  • Undergraduate Algebraic Geometry, by Miles Reid, 1988, Cambridge University Press, London Mathematical Society Student Texts, 12.
    You may use this accessible book as an introduction to the basic ideas about varieties.
  • Algebraic Geometry, by Robin Hartshorne, 1977, Springer, Graduate Texts in Mathematics, 52.
    You may use this book as a technical reference, or for further reading. For example, it covers coherent sheaves, which our text omits.
  • Useful previous courses:

  • Math 321: a whole semester about algebraic geometry, algebraic varieties, and the algorithms that can be used in computations.
  • Math 401: includes three hours on affine schemes.
  • Math 402: includes six hours on algebraic varieties.
  • Math 403: basic theory about the rings that can arise in algebraic geometry
  • Sequel:

    I intend to offer a sequel to this course in Fall, 2001, as Math 416, Topics in Algebra, subtitled Algebraic Geometry, II, and cover coherent sheaves, sheaf cohomology, and the Riemann-Roch theorem, using Hartshorne's book.


    This course will be different enough that if you've taken Math 422 in Spring, 2000, you will be able to get some credit for this course by taking it as a reading course (Math 490) with me.

    Office hours:

    I will hold office hours at times to be announced.

    The grading:

    To be announced.