# ASO: Projective Geometry - Material for the year 2019-2020

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8 lectures

Projective Geometry might be viewed as the geometry of perspective. Two observers of a painting – one looking obliquely, one straight on – will not agree on angles and distances but will both sees lines as lines and will agree on whether they meet. So projective transformations (such as relate the two observers’ views) are less rigid than Euclidean, or even affine, transformations. Projective geometry also introduces the idea of points at infinity – points where parallel lines meet. These points fill in the missing gaps/address some special cases of geometry in a similar way to which complex numbers resolve such problems in algebra. From this point of view ellipses, parabolae and hyperbolae are all projectively equivalent and just happen to include no, one or two points at infinity. The study of such conics also has applications to the study of quadratic Diophantine equations.

Students will be familiar with the idea of projective space and the linear geometry associated to it, including examples of duality and applications to Diophantine equations.

1-2: Projective Spaces (as $P(V)$ of a vector space $V$). Homogeneous Co-ordinates. Linear Subspaces.

3-4: Projective Transformations. General Position. Desargues Theorem. Cross-ratio.

5: Dual Spaces. Duality.

6-7: Symmetric Bilinear Forms. Conics. Singular conics, singular points. Projective equivalence of non-singular conics.

7-8: Correspondence between $P^1$ and a non-singular conic. Simple applications to Diophantine Equations.

1) N.J. Hitchin, *Maths Institute notes on Projective Geometry*, found at http://people.maths.ox.ac.uk/hitchin/hitchinnotes/hitchinnotes.html

2) M. Reid and B. Szendrői, *Geometry and topology*, Cambridge University Press, 2005 (Chapter 5).

3) R. Casse, Projective Geometry, An Introduction, Oxford University Press (2006)