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Origins of generic point in algebraic geometry and mathematical practice

Grant number: 19/01580-7
Support type:Scholarships in Brazil - Master
Effective date (Start): March 01, 2019
Effective date (End): February 28, 2021
Field of knowledge:Humanities - Philosophy
Principal Investigator:Giorgio Venturi
Grantee:Renato Reis Leme
Home Institution: Instituto de Filosofia e Ciências Humanas (IFCH). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
Associated research grant:16/25891-3 - Arbitrariness and genericity: or on how to speak of the unspeakable, AP.JP

Abstract

In Euclidean geometry, the concept of point is a primitive notion characterized intuitively as "that which has no part". Among the concepts defined on the basis of this notion, there is that of scalar: a scalar is what can be described as a point on a scale. Throughout the development of geometry, the notion of point received a number of other characterizations: in analytic geometry, it came to be taken as coordinate, in projective geometry, as projective space, and so on. More recently, with the flourishing of Topology and new developments in algebraic geometry, the concept of point has again been transformed, now on the basis of a movement that Jean Dieudonné, french mathematician and historian of geometry, called "extending the scalars" : from this movement, whose origins this research proposes to ascertain, the concept of generic point could be introduced in the lexicon of algebraic geometers. How, however, to reconcile the necessary individuality of the primitive notion of point with the genericity required by the new concept introduced? Is the generic point concept itself a primitive notion or a concept constructed on the basis of others? In order to offer a historical-conceptual analysis of the development in which culminating the extension of the scalars, we will depart from the scheme of "movements" offered by Dieudonné in 'The historical development of algebraic geometry' (1972), on the basis of which we will defend the hypothesis that the introduction of the concept of generic point was due to the process of increasing abstraction of the objects of study of the algebraic geometry initiated with the formulation of the topology of the varieties of Zariski in the decade of 1940 and culminated in the theory of the schemes of Grothendieck in 1957. On this perspective, we will also evaluate Dieudonné's suggestion that the movement of the scalar extension was a precursor to the method of change of basis of Scheme theory. We will also try to understand how set theory, through the notion of arbitrary sets, could provide a basis for the definition of the concept of generic point. Finally, we expect to find, in Kit Fine's theory of arbitrary objects, a means of understanding the nature of these particular geometric objects which, according to Dieudonné, has reintroduced simplicity to the same extent that it extended the range of the proofs in algebraic geometry.