This chapter is dedicated to the period when projective geometry reached is apex, with the definitive merging of synthetic and analytic-algebraic methods via the use of homogeneous coordinates, and to the transformation it went through in the process towards projective algebraic geometry. It is shown how, in the early 1870s, the appearance of the so-called non-Euclidean geometries and the revolutionary geometrical ideas of Riemann, opened the way to two great questions: to found projective geometry on secure axiomatic bases; and to develop a geometry in spaces of any finite dimension. With this also entered the scene the notions of manifold and 𝑛-dimensional variety, together with the consideration of geometrical transformations more general than the projective ones. Klein’s Erlangen Programme, published in 1872, led to interpret the “new geometries” in term of groups of transformations: each geometry is underlying a specific group of transformations. This view and the debate on von Staudt’s work, in which many mathematicians were involved, contributed to push forward the process of axiomatization which completed itself only in the first decade of 1900. All this is discussed here in a quite detailed manner, together with an overview on those who worked in this process, as Pasch, Peano, Enriques, Fano, and Pieri, among others. In this chapter, following the track that from Cremona and Veronese goes to C. Segre, Bertini, Enriques, Castelnuovo, and Severi, it is told how these mathematicians developed the so-called “Italian school”, and of its influence outside Italy. Finally, it is sketched how the further development of projective algebraic geometry in the twentieth century led to a more and more abstract approach, through the works of Zariski, who studied in Rome in the 1920s, and Weil; and how this, in the 1950s, opened the way to a fair criticism to the methods of the Italian school, no longer suitable for managing the emerging situations. However, in the 1980s, algebraic geometers began to rethink the ties between the “modern” and the “classical” algebraic geometry, and to revaluate the results achieved by the latter. So, the book end by observing that, at almost one century after Coolidge’s address at the Congress of the Mathematical Association of America, mentioned in the Introduction, projective geometry, intended in the broader sense of a geometry in projective hyperspaces has not yet reached the “decline” Coolidge had announced.

Projective Geometry 1870–1930 and Beyond

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发表日期：

2025年

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Andrea Del Centina,
Alessandro Gimigliano

摘要

This chapter is dedicated to the period when projective geometry reached is apex, with the definitive merging of synthetic and analytic-algebraic methods via the use of homogeneous coordinates, and to the transformation it went through in the process towards projective algebraic geometry. It is shown how, in the early 1870s, the appearance of the so-called non-Euclidean geometries and the revolutionary geometrical ideas of Riemann, opened the way to two great questions: to found projective geometry on secure axiomatic bases; and to develop a geometry in spaces of any finite dimension. With this also entered the scene the notions of manifold and 𝑛-dimensional variety, together with the consideration of geometrical transformations more general than the projective ones. Klein’s Erlangen Programme, published in 1872, led to interpret the “new geometries” in term of groups of transformations: each geometry is underlying a specific group of transformations. This view and the debate on von Staudt’s work, in which many mathematicians were involved, contributed to push forward the process of axiomatization which completed itself only in the first decade of 1900. All this is discussed here in a quite detailed manner, together with an overview on those who worked in this process, as Pasch, Peano, Enriques, Fano, and Pieri, among others. In this chapter, following the track that from Cremona and Veronese goes to C. Segre, Bertini, Enriques, Castelnuovo, and Severi, it is told how these mathematicians developed the so-called “Italian school”, and of its influence outside Italy. Finally, it is sketched how the further development of projective algebraic geometry in the twentieth century led to a more and more abstract approach, through the works of Zariski, who studied in Rome in the 1920s, and Weil; and how this, in the 1950s, opened the way to a fair criticism to the methods of the Italian school, no longer suitable for managing the emerging situations. However, in the 1980s, algebraic geometers began to rethink the ties between the “modern” and the “classical” algebraic geometry, and to revaluate the results achieved by the latter. So, the book end by observing that, at almost one century after Coolidge’s address at the Congress of the Mathematical Association of America, mentioned in the Introduction, projective geometry, intended in the broader sense of a geometry in projective hyperspaces has not yet reached the “decline” Coolidge had announced.