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MA30369: Projective geometry

[Page last updated: 22 May 2025]

Academic Year: 2025/26
Owning Department/School: Department of Mathematical Sciences
Credits: 6 [equivalent to 12 CATS credits]
Notional Study Hours: 120
Level: Honours (FHEQ level 6)
Period:
Semester 1
Assessment Summary: EXCB 100%
Assessment Detail:
  • MA32069 Closed-book written examination (EXCB 100%)
Supplementary Assessment:
Like-for-like reassessment (where allowed by programme regulations)
Requisites: Before taking this module you must take MA22020 OR take MA20216
In taking this module you cannot take MA30231
Learning Outcomes: By the end of this unit, you will be able to:
  • Demonstrate knowledge and understanding of the basic properties of projective spaces over arbitrary fields.
  • Work with projective transformations.
  • Demonstrate knowledge and understanding of the geometry of dual spaces, and of quadrics, especially conics.
  • Simultaneously diagonalise conics, and identify when this is possible.



Synopsis: This course will use linear algebra to introduce you to projective geometry, the geometry of perspective. You will strengthen your understanding of linear algebra by exploring its geometrical significance. You will learn about important concepts of projective geometry such as projective linear subspaces, quadrics and cross-ratios and their relationships with both linear algebra and more familiar affine geometry.

Content: Projective spaces over arbitrary fields: projective subspaces, homogeneous and inhomogeneous coordinates, joins and intersections with dimension formula and applications. Projective maps and transformations, points in general position, Desargues' Theorem. Projective lines and cross ratios. Dual projective spaces, annihilators and duality, relation with joins and intersections. Quadrics: bilinear forms and quadratic forms, singular and non-singular quadrics, quadrics on a line, classification of conics, quadric surfaces, polars. Pencils of quadrics, simultaneous diagonalizability and singular quadrics, simultaneous diagonalization for conics.

Course availability:

MA30369 is Optional on the following courses:

Department of Mathematical Sciences
  • USMA-AFM14 : MMath(Hons) Mathematics (Year 4)
  • USMA-AAM15 : MMath(Hons) Mathematics with Study year abroad (Year 4)
  • USMA-AKM15 : MMath(Hons) Mathematics with Year long work placement (Year 5)
Department of Physics
  • USXX-AFM01 : MSci(Hons) Mathematics and Physics (Year 4)
  • USXX-AAM01 : MSci(Hons) Mathematics and Physics with Study year abroad (Year 5)
  • USXX-AKM01 : MSci(Hons) Mathematics and Physics with Year long work placement (Year 5)

Notes:

  • This unit catalogue is applicable for the 2025/26 academic year only. Students continuing their studies into 2026/27 and beyond should not assume that this unit will be available in future years in the format displayed here for 2025/26.
  • 好色tv and units are subject to change in accordance with normal University procedures.
  • Availability of units will be subject to constraints such as staff availability, minimum and maximum group sizes, and timetabling factors as well as a student's ability to meet any pre-requisite rules.
  • Find out more about these and other important University terms and conditions here.