The picture at left shows an affine view of a projective surface which is smooth at all points except those on the curve forming a ridge in the middle of the surface. This surface is the tangent developable to the twisted cubic curve, that is the union of all the tangent lines to the twisted cubic. The projective equation is 3*y^2*z^2-4*x*z^3-4*y^3*w+6*x*y*z*w-x^2*w^2. One can check that the points in projective space for which all partial derivatives vanish are precisely those on the twisted cubic curve. Several lines are drawn on the surface to illustrate how it is swept out by tangent lines to the cubic. We have drawn the real locus in the affine space w=1, clipped to fit inside a spherical region. Such real developable surfaces have the property that they can be assembled out of pieces of sheet metal, since they have zero curvature, generalizing the construction of cylinders or cones by rolling up a sheet. In this example, two flat sheets can be cut such that they are bent and then soldered along the twisted cubic to yield this surface. You can see the tangents as bands of light along the brass interior of the surface.

Mathematics 535: Introduction to Algebraic Geometry, Fall 2008

I last taught this course several years ago. The old course web site gives an idea of what the course covered then.

Description

Summary of the course

References and guide to the literature

Weekly assignments

Each week there will be assigned reading (in the text, other books or journals, or on the internet) and weekly problem sets for which the class will produce solutions. Some readings will be expository or informal.

Solutions to Weekly assignments

We will produce solutions to the problems assigned this semester. See the problem list for a list of the 73 problems assigned this semester and instructions about submitting solutions.

Week 7

• Text Readings: Harris, Chapters 13, 18
• Week 7 Problem set 7A - PDF file (also TeX source for problem statements.)
• Week 7 Problem set 7B - PDF file (also TeX source for problem statements.)
• Week 7 expository reading:
  • Read Oort's note about singularities in the Notices of the AMS.
  • Read the final section of the first chapter of Hartshorne's book on algebraic geometry.
• Algebraic geometry in pictures - Week 7
  • Details about the algebraic geometry movie of the week above.

Week 6

• Text Readings: Harris, Chapters 8, 11
• Week 6 Problem set 6A - PDF file (also TeX source for problem statements.)
• Week 6 Problem set 6B - PDF file (also TeX source for problem statements.)
• Week 6 expository reading:
  • Read a Seminaire Bourbaki report about open problems dealing with affine n-space.
• Algebraic geometry in pictures - Week 6
  • Details about the algebraic geometry movie of the week above.

Week 5

• Text Readings: Harris, Chapters 6, 7
• Week 5 Problem set 5A - PDF file (also TeX source for problem statements.)
• Week 5 Problem set 5B - PDF file (also TeX source for problem statements.)
• Week 5 expository reading:
  • Read Hunt's survey of some aspects of algebraic surfaces.
• Macaulay Worksheet 5.
• Algebraic geometry in pictures - Week 5
  • Details about the algebraic geometry movie of the week above.
  • Cubic Scroll projected to P^3 (blowup of P^1 at a point).

Week 4

• Text Readings: Harris, Chapters 5
• Week 4 Problem set 4A - PDF file (also TeX source for problem statements.)
• Week 4 Problem set 4B - PDF file (also TeX source for problem statements.)
• Week 4 expository reading:
  • Read sections 1-3 of Dereksen and Kraft's survey of invariant theory to understand w hy Hilbert proved the basis theorem and nullstellensatz
• Algebraic geometry in pictures - Week 4
  • Details about the algebraic geometry movie of the week above.
  • Fano variety of a cubic surface.

Week 3

• Text Readings: Harris, Chapters 3,4
• Week 3 Problem set 3A - PDF file (also TeX source for problem statements.)
• Week 3 Problem set 3B - PDF file (also TeX source for problem statements.)
• Week 3 expository reading:
  • Read G. Francis discussion of projections . Though written for topologists it is useful for visualizing the projections of the Veronese surface.
• Macaulay Worksheet 3.
• Algebraic geometry in pictures - Week 3
  • Details about the algebraic geometry movie of the week above.
  • Projections of the twisted cubic.
  • Steiner's interactive Roman Surface .
  • Conics on the roman surface and other projections of the Veronese surface.

Week 2

• Text Readings: Harris, Chapter 2
• Week 2 Problem set 2A - PDF file (also TeX source for problem statements.)
• Week 2 Problem set 2B - PDF file (also TeX source for problem statements.)
• Week2 expository reading:
  • Read sections 1-3 of J. Kollar's article "Which are the simplest algebraic varieties".
• Algebraic geometry in pictures - Week 2
  • Details about the algebraic geometry movie of the week above.
  • Why aren't there any new pictures here of surfaces obtained by cutting a Veronese or Segre variety by a linear space of dimension 3?
  • A curve of type (1,2) on the Segre surfac.

Week 1

• Text Readings: Harris, Chapter 1
• Week 1 Problem set 1A -PDF file ( also TeX source for problem statements.)
• Week 1 Problem set 1B -PDF file ( also TeX source for problem statements.)
• Week1 expository reading:
  • Read D. Freed's expository article on the Hodge Conjecture . If you wish more detail You can find the official description at the prize problem site . To understand it you will need to know about harmonic forms and more, try J. Carlson's exposition of Hodge Theory.
  • H. Poincare wrote "It seems at first sight as if geometry could contain nothing which is not already presented to us in algebra and analytical geometry; for the facts of geometry are nought else than the facts of algebra and analytical geometry expressed in another language." Read section 8 of Poincare's 1908 Rome ICM talk for his opinion of the importance of geometry.
  • Algebraic geometry in pictures - Week 1
    • Details about the algebraic geometry movie of the week above.
    • Pappus Theorem
    • Pascal Theorem
    • Unirationality of the cubic Fermat surface x^3+y^3+z^3+w^3=0.
    • The twisted cubic