Segre and the Foundations of Geometry: From Complex Projective Geometry to Dual Numbers

Abstract

In 1886 Corrado Segre wrote to Felix Klein about his intention to study ‘geometrie projective pure’, completing and developing the work of von Staudt. He would continue this research project throughout the whole of his scientific life. In 1889, following a suggestion of Segre, Mario Pieri published his translation of the Geometrie der Lage, and from 1889 to 1890 Segre published four important papers, “Un nuovo campo di ricerche geometriche”, in which he completely developed complex projective geometry, considering new mathematical objects such as antiprojectivities and studying the Hermitian forms from a geometrical point of view with the related ‘hyperalgebraic varieties’. Segre developed the same ideas in 1892 in a new paper published in the Matematische Annalen, in which he also considers the so-called “bicomplex numbers” and provided the first example of a projective geometry on an algebra with zero-divisors. In 1891, during one of his celebrated courses in higher geometry, Segre asked his students to find a system of independent axioms for projective hyperspatial geometry. Fano, Enriques, Pieri and Amodeo, who wrote important papers, followed this proposal.