# What is a good book to study classical projective geometry for the reader familiar with algebraic geometry?

The more I study algebraic geometry, the more I realize how I should have studied projective geometry in depth before. Not that I don't understand projective space (on the contrary, I am well versed in several different constructions of it), but I lack the familiarity with basic results as cross-ratios, how projective linear transformations act on projective space (as in how many points determine one transformation), Desargues' theorem, etc. I also sometimes feel that it wouldn't hurt to get more practice with hard (as in Olympiad-style) classical geometry problems that may or may not use some facts of projective geometry.

To summarize, I am looking for a reference that covers classical results of projective geometry, and yet assumes the maturity of a reader who has already started studying algebraic geometry. It would be only better if such a book could help me understand where those amazing solutions to Olympiad problems come from.

Does anyone have a suggestion?

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+1 for this extremely original and well-motivated request. – Georges Elencwajg Nov 2 '13 at 17:04
Perhaps just chapter 5 of Reid & Szendroi's Geometry and Topology? – Marius Kempe Feb 20 '15 at 15:46

I) Lectures on Curves, Surfaces and Projective Varieties by Beltrametti, Carletti, Gallarati, Bragadin.
This is a fat textbook written by four Italian geometers in a very classical style and concentrating on classical projective geometry: schemes, cohomology or functors are never even alluded to!

II) Geometria proiettiva, Problemi risolti e richiami di teoria by Fortuna, Frigerio, Pardini.
This is a book consisting of solved exercises preceded by a reminder of the theory.
Although the book is recent the content is very classical and elementary: cross-ratios, quadrics, pencils of conics, inflection points,linear systems,...
From a review:
“This book is the result of the experience acquired by the authors while lecturing Projective Geometry to students from a three year course leading to a degree in Mathematics in the University of Pisa (Italy). … ” (Ana Pereira do Vale, Zentralblatt MATH, Vol. 1227, 2012)
(The book is in Italian, but judging from your first name this might not be a big deal :-)
Anyway mathematical Italian is very close to mathematical English)

I think there is some poetic justice in the fact that all seven authors of the two books are Italian: the justly vaunted Italian algebraic geometry seems to be alive and well in its native country!

Edit
Richter-Gebert has has recently written an encyclopaedic book containing an amazing wealth of material on projective geometry, starting with nine (!) proofs of Pappos's theorem .
The book examines some very unexpected topics like the use of tensor calculus in projective geometry, building on research by computer scientist Jim Blinn .
It would be difficult to read that book from cover to cover but the book is fascinating and has splendid illustrations in color.

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When I google "Lectures on Curves, Surfaces and Projective Varieties" the first item I find is a PDF of the book. Just saying... – Georges Elencwajg Nov 2 '13 at 19:38
Dear Georges, thank you so much for your two suggestions. I'll take a look at them tomorrow, but I think II) is probably more what I'm looking for since it has a concise review of the theory and many exercises. Also I) might be too large for me to make only a "quick break" into classical geometry. – Rodrigo Nov 3 '13 at 6:17
Dear Rodrigo, yes, I) is quite encyclopaedic: I 'm far from knowing all of its content myself... – Georges Elencwajg Nov 3 '13 at 7:00

My recommendations are:

[1] Lynn E. Garner: An Outline of Projective Geometry

[2] A. Seidenberg: Lectures in Projective Geometry

[3] Robin Hartshorne: Foundations of Projective Geometry, http://filebox.vt.edu/users/jabrunso/Math/Hartshorne.pdf

All these books are on classical projective geometry, assuming only basic knowledge. But I think your familiarity with algebraic geometry gives you a higher point of view while reading any of these books. (My personal opinion is that [1] is the best, because it covers the most impotant synthetic results - Desarguesian and Pappian projective planes, projectivities, collineations, polarities, conics, etc. - and the linear algebraic point of view as well.)

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No less a geometer than David Hilbert wrote an accessible book on the subject "Geometry and the Imagination". After finishing Hartshorne's book "Geometry: Euclid and Beyond", I wanted to learn more about configurations and incidence structures. You can find a pdf of Hilbert's book by googling the title, or you can buy a dead tree copy for around thirty bucks.

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Here are some recommendations from my reading list:

1. Linear Geometry by Gruenberg and Weir

2. Projective and Polar Spaces by Peter Cameron

3. Projective Geometry: From Foundations to Applications by Albrecht Beutelspacher and Ute Rosenbaum

4. Outline of Projective Geometry by L. E. Garner

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