Projective geometry is a very classical part of mathematics and one
might think that the subject is completely explored and that there is
nothing new to be added. But it seems that there exists no book on
projective geometry which provides a systematic treatment of morphisms.
We intend to fill this gap. It is in this sense that the present
monograph can be called modern. The reason why morphisms have not been
studied much earlier is probably the fact that they are in general
partial maps between the point sets G and G, noted ' 9: G -- G', i.e.
maps 9: D -4 G' whose domain Dom 9: = D is a subset of G. We give two
simple examples of partial maps which ought to be morphisms. The first
example is purely geometric. Let E, F be complementary subspaces of a
projective geometry G. If x E G \ E, then g(x): = (E V x) n F (where E
V x is the subspace generated by E U {x}) is a unique point of F, i.e.
one obtains a map 9: G \ E -4 F. As special case, if E = {z} is a
singleton and F a hyperplane with z tf. F, then g: G \ {z} -4 F is the
projection with center z of G onto F.