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 Dr V Guletskii
Mathematical Sciences
vladimir.guletskii@liverpool.ac.uk
Year CATS Level Semester CATS Value
Session 2016-17 Level 7 FHEQ Second Semester 15

Aims
  1. 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.
  2. 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.
  3. To understand in detail the proofs of several fundamental results in algebraic geometry on the structure of birational maps and intersection theory. 
  4. To take the first steps in acquiring the technique of linear systems, vector bundles and differential forms.

Learning Outcomes

To know:basic concepts of smooth geometry and algebraic geometry.

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.

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

Syllabus
1 
  • 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.
Explanation of Reading List:

Pre-requisites before taking this module (other modules and/or general educational/academic requirements):
MATH101; MATH142; MATH244; MATH247; MATH248; MATH102; MATH103  

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:
Programme:G101 Year:3,4 Programme:MMAS Year:1

Assessment
EXAM Duration Timing
(Semester)		 % of
final
mark		 Resit/resubmission
opportunity		 Penalty for late
submission		 Notes
Unseen Written Exam  180  Semester 2  90  Yes    Written exam Notes (applying to all assessments) Coursework is assessed through weekly homeworks.  
CONTINUOUS Duration Timing
(Semester)		 % of
final
mark		 Resit/resubmission
opportunity		 Penalty for late
submission		 Notes
Coursework    2  10  Yes  Standard UoL penalty applies  Weekly homework