MATH 431
Algebraic Geometry
Spring 2022
Ali Sinan Sertöz
Faculty of Science, SA-Building, Department of Mathematics, Room 121
Phone: 1490
Schedule:
TUE 13:30-15:20 SA-Z03
FRI 08:30-10:20 SA-Z03
General Notes:
I have given several Math 431 courses in the past, and no two were alike!
You may browse a list of these courses at this link.
This time I plan to merge the algebraic and complex analytic aspects of geometry as follows:
The beginnings of algebraic geometry lie in the theory of Riemann Surfaces which are complex curves. Most of the powerful tools of higher dimensional algebraic geometry take their inspiration from what Riemann did with curves.
We will thus combine the general tools of algebraic geometry with their applications to the specific subject of complex curves.
One of the mind blowing results of complex curve theory is that the abstractly defined compact complex Riemann surfaces can all be realized as concrete algebraic varieties which are solution sets of polynomials only, no other complex analytic functions are involved contrary to what we were expecting.
To appreciate the staggering beauty of such results we must develop a language which will both help us to talk about such objects without ambiguity and also help us to understand why such results are unexpected, beautiful and exciting.
Main Textbook: Robin Hartshorne, Algebraic Geometry, Springer-Verlag, 1977.
Auxiliary Textbook: Phillip Griffiths, Introduction to Algebraic Curves, American Mathematical Society, 1989
From the first book we will learn the general techniques and approach and from the second book we will see how these ideas apply to curves.
Highlights of the course will include:
FZ Requirement: Less than 50% attendance and/or less than 35% average from the midterms may result in an FZ grade.
(Statistically we know that students whose midterm averages are low never write a good final exam paper which will make their letter grade above F.)
Exams and Homework
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Faculty of Science, SA-Building, Department of Mathematics, Room 121
Phone: 1490
Schedule:
TUE 13:30-15:20 SA-Z03
FRI 08:30-10:20 SA-Z03
General Notes:
I have given several Math 431 courses in the past, and no two were alike!
You may browse a list of these courses at this link.
This time I plan to merge the algebraic and complex analytic aspects of geometry as follows:
The beginnings of algebraic geometry lie in the theory of Riemann Surfaces which are complex curves. Most of the powerful tools of higher dimensional algebraic geometry take their inspiration from what Riemann did with curves.
We will thus combine the general tools of algebraic geometry with their applications to the specific subject of complex curves.
One of the mind blowing results of complex curve theory is that the abstractly defined compact complex Riemann surfaces can all be realized as concrete algebraic varieties which are solution sets of polynomials only, no other complex analytic functions are involved contrary to what we were expecting.
To appreciate the staggering beauty of such results we must develop a language which will both help us to talk about such objects without ambiguity and also help us to understand why such results are unexpected, beautiful and exciting.
Main Textbook: Robin Hartshorne, Algebraic Geometry, Springer-Verlag, 1977.
Auxiliary Textbook: Phillip Griffiths, Introduction to Algebraic Curves, American Mathematical Society, 1989
From the first book we will learn the general techniques and approach and from the second book we will see how these ideas apply to curves.
Highlights of the course will include:
- Bezout's Theorem: Two plane curves of degrees m and n intersect in exactly mn points. If you see less than that many points then some of the intersection points are at infinity. We need to develop a language to control what happens at infinity. That is why we will first develop a theory of projective spaces.
- Riemann-Roch Theorem: You can control the number of functions on a curve and thus find enough number of functions to embed a curve into a projective space. This is totally unexpected but true! And once you are in projective space then you have only polynomials!! This is due to a deep theorem of Serre which goes by the funny name GAGA.
- On a surface of degree three there are exactly 27
lines, no matter which cubic surface you take. But on
a generic surface of higher degree there are no lines at
all. Some surfaces however have some lines and the possible
number of lines a surface can hold is an on going active
research area. (see for example this paper.)
- Time permitting I will briefly introduce the ideas around
Arf rings and Arf closures which are related
to the understanding of singular, bad, points of a curve. (see for example
this paper.)
I will mainly follow the syllabus given on STARS with occasional deviations to
cover more interesting topics.
Grading: I will follow the midterm and homework
requirements as announced in STARS.
Letter Grades:
The course will be graded according to the following catalogue:
| [0,34) | F |
| [34,40) | D |
| [40,44) | D+ |
| [44,50) | C- |
| [50,55) | C |
| [55,59) | C+ |
| [59,63) | B- |
| [63,65) | B |
| [65,70) | B+ |
| [70,75) | A- |
| [75,100] | A |
FZ Requirement: Less than 50% attendance and/or less than 35% average from the midterms may result in an FZ grade.
(Statistically we know that students whose midterm averages are low never write a good final exam paper which will make their letter grade above F.)
Exams and Homework
| Midterm 1 |
30% |
01 April 2022 Friday |
Solution
|
| Midterm 2 |
30% |
06 May 2022 Friday |
Solution |
| Homework
1 |
1% |
04 March 2022 |
Solution |
| Homework
2 |
1% |
08 April 2022 Friday |
Solution |
| Homework
3 |
1% | 15 April 2022 Friday | Solution |
| Homework
4 |
1% | 22 April 2022 Friday | Solution |
| Homework
5 |
1% | 29 April 2022 Friday | Solution |
| Final |
35% |
20 May 2022 Friday |
Solution |
| Syllabus | ||||
| Week |
Date |
Hours | Subjects to be covered | Book |
| 1 | 1-4 Feb |
3 | Review of some fundamental concepts from
topology and abstract algebra |
Hartshorne |
| 2 | 8-11 Feb |
3 | Affine spaces, Zariski topology,
Geometry-Algebra dictionary, irreducibility,
coordinate ring of an affine variety |
Hartshorne |
| 3 | 15-18 Feb |
3 | Coordinate rings of affine varieties,
morphisms, equivalence of the category of affine
varieties and polynomial maps with the category of
commutative reduced rings with unity and ring
homomorphisms. |
Hartshorne |
| 4 | 22-25 Feb |
3 | Projective space and projective
varieties |
Hartshorne |
| 5 | 1-4 Mar |
3 | Acceptable morphisms |
Hartshorne |
| 6 | 8 Mar |
2 | Regular rings |
Hartshorne |
| 7 | 15-18 Mar |
3 | Singularities |
Hartshorne |
| 8 | 22-25 Mar |
3 | Curve singularities and Arf Rings |
Hartshorne |
| 9 | 29 Mar-1 Apr |
3 | Intersections in projective space |
Hartshorne |
| 10 | 5-8 Apr |
3 | Problems |
Hartshorne |
| 11 | 12-15 Apr |
3 | Fundamentals of Compact Riemann Surfaces |
Griffiths |
| 12 | 19-22 Apr |
3 | Riemann-Hurwitz formula, degree-genus
formula, divisors. |
Griffiths |
| 13 | 26-29 Apr |
3 | Riemann-Roch Theorem |
Griffiths |
| 14 | 6 May |
1 | Canonical curves |
Griffiths |
| 15 | 10-13 May |
3 | Hyperelliptic and nonhyperelliptic
curves |
Griffiths |
| Total class hours: |
42 |
Griffiths | ||
Write me if you have any questions about
this course.
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