The picture at left cycles through the different nondegenerate inequivalent projections of the Veronese surface in P^5 to 3-space. The real locus of an affine open subset of the projective variety is pictured, with the green lines marking the singular locus, where the gradient of the defining equation is 0. These are Steiner surfaces, each parameterized by quadratic polynomials in 2 variables. All except the ellipsoid are topologically immersions of the real projective plane in real 3-space.

Mathematics 535: Introduction to Algebraic Geometry, Fall 2002

Description

References and guide to the literature

Weekly assignments

Week 3

• Text Readings: Harris, Chapters 3,4
• Week 3 Problem set 3 - 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 2 - 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".
• Macaulay Worksheet 2.
• 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 surface.

Week 1

• Text Readings: Harris, Chapter 1
• Week 1 Problem set 1 -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.
• Macaulay Worksheet 1. Read the introduction to Macaulay 2 at Rutgers if you have not used Macaulay 2 before.
• Algebraic geometry in pictures - Week 1
  • Details about the algebraic geometry movie of the week 1.
  • Pappus Theorem
  • Pascal Theorem
  • Unirationality of the cubic Fermat surface x^3+y^3+z^3+w^3=0.
  • The twisted cubic