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 automorphsim group isomorphic to the Weyl group of the excepitonal 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 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.

The picture at left shows a linear system of divisors on the affine curve y^2=x(x-2)(x+2) which is an affine part of a degree 3 projective smooth plane curve X obtained by removing a point P on the line at infinity. As the blue line intersects the cubic curve the three points of intersection form an effective divisor on X. This is not a complete linear system, since all lines pass through the point (-1,-1). This is a linear system of projective dimension 1 in the 2 dimensional complete linear sysem associated to the divisor 3P. Sometimes the third point of the effective divisor is outside the frame of the picture. Using the sections correponding to these effective divisors gives a morphism of the curve X to the projective line.

Mathematics 536: Introduction to Algebraic Geometry II, Spring 2003

Description

References and guide to the literature

Weekly assignments

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.