The picture at left shows the 28 bitangent lines to the red quartic curve with real locus consisting of 4 ovals. Each line is tangent to the curve at 2 points. The bitangents are colored so that rotation by 90 degrees maps a line to one of the same color. This curve is a smooth plane quartic, so it has genus 3. It is a theorem of Harnack that at most g+1 ovals are possible in the real locus of a plane curve. The configuration of lines has automorphism group isomorphic to the Weyl group of the exceptional Lie algebra E7. Every plane projective quartic has 28 bitangents.

Mathematics 536: Introduction to Algebraic Geometry II, Spring 2003

Description

References and guide to the literature

Weekly assignments

Week 7

• Text Readings: Hartshorne, IV.3,IV.5
• Week 7 Problem set 7 - PDF file (also TeX source for problem statements.)
• Week 7 expository reading:
  • Read about the history of the geometry of a genus 3 curve, the Klein quartic. You will need the following figure 1 missing from the pdf file,
• Macaulay Worksheet 7.
• Algebraic geometry in pictures - Week 7
  • Details about the algebraic geometry movie of the week 7.

Week 6

• Text Readings: Hartshorne, IV.1-IV.3
• Week 6 Problem set 6 - PDF file (also TeX source for problem statements.)
• Week 6 expository reading:
  • Read about the history of the Riemann-Roch theorem.
• Macaulay Worksheet 6.
• Algebraic geometry in pictures - Week 6
  • Details about the algebraic geometry movie of the week 6.

Week 5

• Text Readings: Hartshorne, II.7 (through 7.8.6) , II.8
• Week 5 Problem set 5 - PDF file (also TeX source for problem statements.)
• Week 5 expository reading:
• Macaulay Worksheet 5. THERE IS ACTUALLY A WORKSHEET THIS WEEK!
• Algebraic geometry in pictures - Week 5
  • Details about the algebraic geometry movie of the week 5.

Week 4

• Text Readings: Hartshorne, II.5,6
• Week 4 Problem set 4 - PDF file (also TeX source for problem statements.)
• Week 4 expository reading:
• Macaulay Worksheet 4.
• Algebraic geometry in pictures - Week 4
  • Details about the algebraic geometry movie of the week 4.

Week 3

• Text Readings: Hartshorne, II.4
• Week 3 Problem set 3 - PDF file (also TeX source for problem statements.)
• Week 3 expository reading:
• Macaulay Worksheet 3.
• Algebraic geometry in pictures - Week 3
  • Details about the algebraic geometry movie of the week 3.

Week 2

• Text Readings: Hartshorne, II.2 II.3
• Week 2 Problem set 2 - PDF file (also TeX source for problem statements.)
• Week2 expository reading:
  • Read a short introduction to toric varieties .
  • Read about Grothendieck's work , especially sections 3-5 dealing with the development of the notion of a scheme.
• Macaulay Worksheet 2.
• Algebraic geometry in pictures - Week 2
  • Details about the algebraic geometry movie of the week 2.

Week 1

• Text Readings: Hartshorne II.1, II.2
• Week 1 Problem set 1 -PDF file ( also TeX source for problem statements.)
• Week1 expository reading:
  • Read a book review that addresses the question "How much commutative algebra does an algebraic geometer need to know".
  • Read a review of the book on schemes by Eisenbud and Harris.
• Macaulay Worksheet 1.
• Algebraic geometry in pictures - Week 1
  • Details about the algebraic geometry movie of the week 1.