Math 633
Algebraic Geometry

Fall 2019



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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: Hartshorne, Algebraic Geometry
Supplementary textbook: Görtz and Wedhorn, Algebraic Geometry-I
Both 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.