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). The authors recognize the existence of modern textbooks on this subject published in Italian but felt the need of a different type of book. In fact this book is mainly designed to accompany the student in the learning process. It is not to be read as a treaty on the subject. It is divided into two parts. The first part (pages 1-60) deals with the notation and recalls the main theorems and concepts of projective geometry. It contains nine chapters dealing with: Projective spaces and subspaces and projective transformations (1.2); projective frames and homogeneous coordinates (1.3); dual projective space and duality (1.4); projective spaces of dimension 1 (1.5); conjugation and complexification (1.6); affine and projective hypersurfaces (1.7); quadrics (1.8); and plane algebraic curves (1.9). I.e. the first part of the book contains some fundamental results of projective geometry without proofs. According to the authors this part of the book aims to give the reader a more global idea of the subject and to introduce the notation and concepts used afterwards. The second part of the book (the next 200 pages) include three different chapters and present the solutions for different types of exercises. Exercises on projective spaces, exercises on curves and hypersurfaces and exercises on conics and quadrics. The authors do not prefer the analytic or the synthetic method. The solution given here is the one the authors consider the most interesting, the most elegant or the fastest regardless of the method. In many exercises there is more than one solution. In some exercises as many as three different solutions are given. The authors use an interesting notation to call the reader’s attention to some of the problems given in the book. More difficult exercises are marked with a cup which means according to the authors “mettiti comodo, prenditi un caffé o un té, armati di pazienza e determinazione, e vedrai che ne verrai a capo” (sit comfortably, have a cup of coffee or tea, arm yourself with patience and determination and you will see it through). In some exercises the authors show an alternative solution that is more elaborated but leads to a deeper understanding of the subject. This type of solutions is also signaled. This is an impressive collection of very interesting exercises: 52 for projective spaces, 67 for curves and hypersurfaces and 87 for conics and quadrics.
Reviewer: Ana Pereira do Vale (Braga)