The Early Historical Research On Birational Transformation

The basic object of study in algebraic geometry is the algebraic variety,and the birational transformation(also known as Cremona transformation)is an important tool for classifying algebraic variety.The birational transformation is an important part of the development of algebraic geometry,and its introduction had made algebraic geometry a true discipline,providing powerful tools for many subsequent fields of study.The germ of the idea of the birational transformation can be traced back to the Greek’ Apollonius,but it wasn’t explicitly stated by Riemann until the 19 th century.Later,Cremona established the theory of general birational transformation,which was developed by Max Noether and Bertini,et al.Tracing the origins of the birational transformation can not only help people to understand the motivation,connotation,development process and impact of the birational transformation,but also help people to grasp its application in modern mathematics.Therefore,it is of great theoretical value and practical importance to study the early history of the birational transformation.Based on a large number of original and research literature in German and Italian,this dissertation focuses on the motivation,ideological connotations,and development of the birational transformation and its role in algebraic geometry from the 1830 s to the early 20 th century,comprehensively using the research methods of chronicle,conceptual analysis,documentary research,sociological and other research methods.The main research results and conclusions of this dissertation are as follows:1.The germ of the idea of the birational transformation and the inverse transformation of a circle are explored.The idea of the birational transformation can be traced back to the 5th century BC when Apollonius first described the inverse transformation of a circle in his“Plane loci”.However,the idea of inverse transformation was not further developed for the next two thousand years.It was not until the 1830 s that M(?)bius and Plücker independently introduced homogeneous coordinates to coordinate the elements at infinity,which provided a complete tool for the study of transformation theory.Then,a new revival of the theory of the inverse transformation began.In 1855,M(?)bius developed the theory of circle transformation of the plane,that is,the inverse transformation of a circle,in his paper “Die theorie der kreisverwandtschaft in rein geometrischer darstellung”.2.The proposal of the birational transformation is discussed.In 1857,Riemann studied Abelian functions,Abelian integrals and the inverse of Abelian integrals in his paper “Theorie der Abel’schen functionen”,explicitly proposed the birational transformation for the first time,initiated the study of the birational geometry of algebraic curves,and took the first step in the study of the birational transformation of curves.After that,Riemann’s work was developed by Clebsch.3.The ideological origin,process of proposing and influence of Cremona’s theory of general birational transformation are analyzed.In 1832,Magnus came to the erroneous conclusion in his article “Nouvelle méthode pour découvrir des théorgrave mes de géométrie”that the order of the first class transformation could not be greater than 2,and implied in the footnote that two quadratic transformations could be compounded,which provided Cremona with the key idea,and he thus for concluded that “it is obviously possible to obtain higher order first class transformation by the compounding of quadratic transformations”,which in turn provided the basis for Cremona to expose and correct the above erroneous conclusion.Thus,Magnus’ work provided a direct source of ideas for Cremona.In 1863,in his paper“Sulle trasformazioni geometriche delle figure piane.Note I”,Cremona systematically described the birational geometric transformation of plane figures,which later became known as the Cremona transformation.Two years later,Cremona studied the birational transformation in more detail,proving that the inverse of a birational transformation was also a birational rational,and identifying all the special birational transformations for n(27)8.Cremona’s work established the main outlines for the subsequent development of the theory of the birational transformation.4.The contribution of Max Noether and Bertini to the birational transformation is analyzed.In 1870,Max Noether proved a fundamental result that the planar birational transformation could be composed of a series of quadratic and linear transformations.In 1877,Bertini classified the planar Cremona transformation.In 1880,Bertini proved two theorems that now bear his name: Bertini’s theorems on variable singular points and reducible linear systems.They reached many important conclusions and properties,which have promoted the rapid development of the birational transformation.5.A classical application of the birational transformation is described,i.e.the resolution of a singularity.From the singularities of algebraic curves to the singularities of present scheme,this problem has always been an important problem in algebraic geometry.Among them,the most classic is the problem of resoluting the singularity of algebraic curves.