MATH40870 Algebraic Geometry

Academic Year 2024/2025

First, notions from commutative algebra are reconsidered from a geometric viewpoint. Affine varieties are introduced and endowed with a natural topology (Zariski topology) as well as a natural class of functions (regular functions). This also provides a more geometric perspective on the localization of commutative rings.

Second, projective and quasi-projective varieties are studied. These can be obtained from affine varieties through a gluing process. This gluing is an algebraic analogue of the way manifolds are constructed from local charts. If time allows, we introduce prevarieties and varieties in full generality.

Third, the notion of dimension in algebraic geometry is introduced. If time permits, we prove the semicontinuity of fiber dimensions. Additionally, we introduce Poincaré series and Hilbert polynomials to facilitate the introduction and comparison of various notions of dimension in algebraic geometry.

Finally, we delve into the local geometry of varieties by introducing tangent spaces and cones. In particular, we examine smooth points of varieties and establish a Jacobian criterion.

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Curricular information is subject to change

Learning Outcomes:

Upon successful completion of this module, students

(1) can state and apply the fundamental definitions and results of algebraic geometry in the setting of quasi-projective varieties,

(2) can interpret results from commutative algebra from a geometric perspective,

(3) are able to investigate the geometry of given explicit examples of quasi-projective varieties,

(4) are able to appreciate the modern terminology of algebraic geometry (scheme theory) to be encountered in graduate studies.

Student Effort Hours: 
Student Effort Type Hours
Lectures

30
Tutorial

6
Autonomous Student Learning

84
Total

120
Approaches to Teaching and Learning:
lectures; enquiry & problem-based learning in homework sheets and tutorials 
Requirements, Exclusions and Recommendations

Not applicable to this module.


Module Requisites and Incompatibles
Not applicable to this module.
 
Assessment Strategy  
Description Timing Component Scale Must Pass Component % of Final Grade In Module Component Repeat Offered
Exam (In-person): Final Exam End of trimester
Duration:
2 hr(s)
 Standard conversion grade scale 40% No

70

 No
Exam (In-person): Midterm Exam Week 7 Standard conversion grade scale 40% No

30

 No

Carry forward of passed components
Yes
 
Resit In Terminal Exam
Autumn Yes - 2 Hour
Please see Student Jargon Buster for more information about remediation types and timing. 
Feedback Strategy/Strategies

• Feedback individually to students, post-assessment
• Group/class feedback, post-assessment

How will my Feedback be Delivered?

Not yet recorded.

We mainly follow:
(1) Flaminio Flamini; First Course In Algebraic Geometry And Algebraic Varieties, World Scientific, 2023

Further textbooks that may be used on occasion and by reference:
(2) Thomas Garrity et al, Algebraic Geometry: A Problem Solving Approach, AMS, 2013
(3) Klaus Hulek, Elementary Algebraic Geometry, AMS, 2003
(4) Daniel Perrin, Algebraic Geometry: An Introduction, Springer, 2008
(5) Miles Reid, Undergraduate Algebraic Geometry, Cambridge University Press, 2010