### Projective Geometry (MATH 526)

2021 Fall
Faculty of Engineering and Natural Sciences
Mathematics(MATH)
3
10.00
Michel Lavrauw -mlavrauw@sabanciuniv.edu,
English
Doctoral, Master
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### CONTENT

Homogeneous coordinates, projective spaces, the principle of duality, projective planes and the configurations of Desargues and Pappus, collineations and correlations, perspectivities, the projective groups, polarities, algebraic varieties, classical polar spaces, Plücker coordinates, the Klein quadric, Segre varieties, Veronese varieties.

### LEARNING OUTCOME

By the end of the course the students

1. should be able to work with homogeneous coordinates, quotient spaces, and projections in projective spaces,

2. explain and apply the principle of duality, and reproduce the definitions of collineations, correlations, elations, homologies, perspectivities,

3. should be able to state and prove the Fundamental Theorem of Projective Geometry,

4. should be able to work with the action of the classical groups on projective and polar spaces,

5. should be able to state and prove the classical theorems of Desargues and Pappus for projective planes over fields,

6. should be able to give examples non-Desarguesian projective planes,

7. should be able to state and prove combinatorial properties for finite projective planes,

8. should be able reproduce the coordinatization method for projective planes, explain the connection between the algebraic structures of ternary rings (in particular quasifields, semifields, nearfields, division rings, skewfields) and the geometric and group theoretic properties of the corresponding projective planes (in particular, the different types of translation planes),

9. should also be able to define the basic concepts related to the study of algebraic varieties in affine and projective spaces, give the classification of sesquilinear and quadratic forms and the connection with quadratic and hermitian varieties in projective spaces as well as their relation with the classical groups; define the Klein correspondence and the Klein quadric, the Segre variety and the Veronese variety.