Module Details

The information contained in this module specification was correct at the time of publication but may be subject to change, either during the session because of unforeseen circumstances, or following review of the module at the end of the session. Queries about the module should be directed to the member of staff with responsibility for the module.
Title Algebraic Geometry
Code MATH448
Coordinator Professor AV Pukhlikov
Mathematical Sciences
Pukh@liverpool.ac.uk
Year CATS Level Semester CATS Value
Session 2020-21 Level 7 FHEQ Second Semester 15

Aims

To give a detailed explanation of basic concepts and methods of algebraic geometry in terms of coordinates and polynomial algebra, supported by strong geometrical  intuition.

To elaborate examples and to explain the basic constructions of algebraic geometry, such as projections, products, blowing up, intersection multiplicities, linear systems, vector bundles, etc.

To understand in detail the proofs of several fundamental results in algebraic geometry on the structure of birational maps and intersection theory.

To take the first steps in acquiring the technique of linear systems, vector bundles and differential forms.


Learning Outcomes

(LO1) To know:basic concepts of smooth geometry and algebraic geometry.

(LO2) To understand: the interplay between local and global geometry, the duality between differential forms and submanifolds, between curves and divisors, between algebraic and geometric data.

(LO3) To be able to: perform elementary computations with differential forms, patch together local objects into global ones, compute elementary intersection indices.

(S1) Problem solving skills


Syllabus

 

Affine spaces and Hilbert’s Nullstellensatz.
Homogeneous ideals and geometry in projective spaces.
Height of ideals and Krull dimension.
Regular morphisms and rational maps, sheaves of regular functions, birational equivalence of algebraic varieties.
The Segre map and products of algebraic varieties.
The local ring at a point, tangent spaces, local parameters and Taylor expansions.
Blow up and the structure of birational maps.
Intersection multiplicities and the Bezout theorem.
Vector bundles and locally free sheaves.
Linear systems and line bundles.
Tangent and cotangent bundles, differential forms, the canonical sheaf.


Recommended Texts

Reading lists are managed at readinglists.liverpool.ac.uk. Click here to access the reading lists for this module.

Pre-requisites before taking this module (other modules and/or general educational/academic requirements):

MATH101 CALCULUS I; MATH102 CALCULUS II; MATH101 CALCULUS I; MATH102 CALCULUS II; MATH244 LINEAR ALGEBRA AND GEOMETRY; MATH247 COMMUTATIVE ALGEBRA; MATH248 GEOMETRY OF CURVES; MATH244 Linear Algebra and Geometry; MATH247 MATH247 - Commutative Algebra; MATH248 Geometry of Curves; MATH103 INTRODUCTION TO LINEAR ALGEBRA; MATH103 INTRODUCTION TO LINEAR ALGEBRA 

Co-requisite modules:

 

Modules for which this module is a pre-requisite:

 

Programme(s) (including Year of Study) to which this module is available on a required basis:

 

Programme(s) (including Year of Study) to which this module is available on an optional basis:

 

Assessment

EXAM Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
final assessment open book and remote  1 hour time on task    50       
CONTINUOUS Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Homework 3  equivalent to 2-5 si    10       
Homework 2  equivalent to 2-5 si    10       
Homework 1  equivalent to 2-5 si    10       
Homework 5  equivalent to 2-5 si    10       
Homework 4  equivalent to 2-5 si    10