Math 633
Algebraic Geometry
Fall 2019
.
Algebra and Geometry
Ali
Sinan Sertöz
Faculty of Science A-Block room 121
Phone: 290 1490
Office Hours: anytime, in person or via email, 7/24
Schedule:
TUE: 13:40-15:30 B-108
THU: 15:40-16:30 B-108
Spare Hour: THU: 16:40-17:30 B-108
Main Textbook:
1) Hartshorne, Algebraic Geometry
Supplementary textbooks:
2) Görtz and Wedhorn, Algebraic Geometry-I
3) Griffiths and Harris, Principles of Algebraic Geometry
**All of these books can be downloaded in pdf form if you are connected to Bilkent network.**
Grading will be according to Homework and Take-Home problem solutions.
Syllabus:
Since I intend to follow the pace of the students in the class I will not write a timetable for the syllabus.
Our goal is to understand the basic concepts of algebraic geometry as treated in the first chapter of Hartshorne's book such as affine and projective varieties, maps between these objects and in particular the blow-up map. We will examine algebraic invariants of our geometric objects such as the ring of all functions on them, globally or locally defined.
With these geometric intuition behind us we will embark into the next chapter in Hartshorne and start studying the modern language of algebraic geometry, namely sheaves and schemes. Here we may from time to time consult the supplementary textbook mentioned above.
For the record here is a tentative list of the topics we intend to cover:
Review: topology for open/closed sets, Krull dimension of a topological space, polynomial ring, ideals, in particular prime and maximal ideals, Noetherian condition both for rings and topological spaces. (We will review category theoretical concepts as they appear during our study.)
Affine varieties: Affine space, closed subsets, varieties, dimension, morphisms, local and global functions
Projective varieties: Projective space, closed subsets, dimension, morphisms, local and global functions, projective closure of an affine variety.
Sheaves and schemes: A scheme is basically a collection of data providing in one definition all the properties of a geometric object. This necessarily involves non-intuitive definitions and some technical machinery which we will study at our own pace.
Faculty of Science A-Block room 121
Phone: 290 1490
Office Hours: anytime, in person or via email, 7/24
Schedule:
TUE: 13:40-15:30 B-108
THU: 15:40-16:30 B-108
Spare Hour: THU: 16:40-17:30 B-108
Main Textbook:
1) Hartshorne, Algebraic Geometry
Supplementary textbooks:
2) Görtz and Wedhorn, Algebraic Geometry-I
3) Griffiths and Harris, Principles of Algebraic Geometry
**All of these books can be downloaded in pdf form if you are connected to Bilkent network.**
Grading will be according to Homework and Take-Home problem solutions.
Syllabus:
Since I intend to follow the pace of the students in the class I will not write a timetable for the syllabus.
Our goal is to understand the basic concepts of algebraic geometry as treated in the first chapter of Hartshorne's book such as affine and projective varieties, maps between these objects and in particular the blow-up map. We will examine algebraic invariants of our geometric objects such as the ring of all functions on them, globally or locally defined.
With these geometric intuition behind us we will embark into the next chapter in Hartshorne and start studying the modern language of algebraic geometry, namely sheaves and schemes. Here we may from time to time consult the supplementary textbook mentioned above.
For the record here is a tentative list of the topics we intend to cover:
Review: topology for open/closed sets, Krull dimension of a topological space, polynomial ring, ideals, in particular prime and maximal ideals, Noetherian condition both for rings and topological spaces. (We will review category theoretical concepts as they appear during our study.)
Affine varieties: Affine space, closed subsets, varieties, dimension, morphisms, local and global functions
Projective varieties: Projective space, closed subsets, dimension, morphisms, local and global functions, projective closure of an affine variety.
Sheaves and schemes: A scheme is basically a collection of data providing in one definition all the properties of a geometric object. This necessarily involves non-intuitive definitions and some technical machinery which we will study at our own pace.
| Syllabus | |
| Week |
-----------------------------------------Topics Covered----------------------------------------- |
| 1 |
Topological Spaces: Open, closed,
irreducible, dimension, examples |
| 2 |
Affine Varieties: Polynomial ring $k[x_1,\dots,x_n]$, ideals, zero sets of ideals, ideal of subsets of $\mathbb{A}^n$, Zariski topology on $\mathbb{A}^n$ |
| 3 |
Functions on affine varieties:
coordinate ring, local ring, regular and rational
functions, morphism between varieties and morphisms
between their coordinate rings |
| 4 |
Smooth vs Singular: analytic
definition of smoothness and algebraic counterpart,
regular local rings |
| 5 |
Projective varieties: graded ring
$k[x_0,\dots,x_n]$, homogeneous ideals, projective
algebraic sets, Zariski topology on $\mathbb{P}^n$,
projective coordinate ring. |
| 6 |
Quasi-projective varieties:embedding
affine varieties into projective space, projective
closure, products of varieties, intersection in
projective space |
| 7 |
Blow-up map:construction of
blow-up space, local charts, resolution of singularities
of a singular curve |
| 8 |
Arf Rings:multiplicity sequence,
blow-up, du Val characters, Arf rings and Arf closure,
arithmetic of Arf rings |
| 9 |
Grassmannians: affine affine
charts, Schuberts cells, homology, Schubert calculus,
Plücker embedding, incidence variety and its dimension |
| 10 |
Presheaves and sheaves: morphisms,
kernel, cokernel, image, sheafification, direct and
inverse images |
| 11 |
Schemes: Spectrum of $A$, ringed
spaces and morphisms |
| 12 |
Affine schemes: patching
spectrums, affine line, affine plane, generic point,
affine line with a double point |
| 13 |
Projective schemes: $Proj$ of a
grades ring, schemes over a base scheme |
| 14 |
First properties of schemes:topological
properties, open and closed schemes, sheaves of modules,
Weil and Cartier divisors, invertible sheaves |
| 15 |
Cech cohomology |
A Summary of what we have covered:
In 1978, in the opening paragraph of Introduction to Commutative Algebra and Algebraic Geometry, Ernst Kunz writes:It has been estimated that, at the present state of our knowledge, one could give a 200 semester course on commutative algebra and algebraic geometry without ever repeating himself. So any introduction to this subject must be highly selective.Our selection in 2019 is intended to define the main objects of study in algebraic geometry and their refinements, and give some immediate applications.
- We started with studying the basic
case of the affine varieties and the maps between
them. We then studied projective varieties. Since
affine varieties can be mapped as open subsets into
projective spaces, we agreed to include open subsets
of projective varieties into our collection of objects
of study.
- Then we took a break from this
theoretical development and used our background to
study two interesting topics.
- One is the blow-up map which is the
main tool to understand singularities. And to
demonstrate its effective use we studied the
geometric aspects of Arf rings, coming from curve
singularities.
- The other application we studied is
the Grassmannian of $k$ planes in an $n$-dimensional
vector space. We investigated its main properties
and as an application of the techniques developed
from Grassmannians we proved that a generic smooth
complex surface in $\mathbb{P}^3$ contains no lines.
Then we mentioned the recent results on determining
the number of lines on special surfaces, which has
recently become an active research topic. As another application of Grassmannian
techniques we attacked some counting problems using
the Schubert cycles of Grassmannians. As an
exemplary demonstration of the effectiveness of
Schubert techniques we calculated that there are
only two lines intersecting four generic lines in
$\mathbb{P}^3$.
- Then we proceeded to learn the
language of modern algebraic geometry, namely sheaves
and schemes. This part is aimed to develop a new
language. Hence inevitably interesting theorems are
rare at the beginning. Here we again call in a defense
witness! In the introduction of his legendary book The
Red Book of Varieties and Schemes, which is 300+
pages, David Mumford writes:
The weakness of these notes is what had originally driven me to undertake the bigger project: there is no real theorem in them!
(Here the bigger project that Mumford refers to, in 1988, is the two volume book on an introduction to algebraic geometry, the second volume of which has appeared only in 2015 with the collaboration of Tadao Oda.)
Nonetheless we ended our discussions with an actual calculation of the divisor class of a quadric surface and its relation to the twisted cubic.
- To conclude the course with something
concrete we develop the theory of Cech cohomology and
actually calculate the cohomology of some familiar
spaces using their structure sheaves.
| Exam |
Due Date |
Solution |
| Exam01 |
5 November 2019 |
Solution |
| Exam02 |
7 November 2019 |
Solution |
| Exam03 |
5 December 2019 |
Solution |
| Exam04 |
5 December 2019 |
Solution |
| Exam05 |
12 December 2019 |
Solution |
| Exam06 |
31 December 2019 |
Solution |
| Final Exam |
10 January 2020 |
Solution |