# .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.

 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