The Riemann-Roch Theorem : A Hundred Years of Algebra

The theme of this book is the study of algebraic curves, moving quickly from the origins of this topic in the work of Descartes and its gradual ramification in the 18th century, to the definitive reformulation of the mid-19th century in terms of complex projective geometry.

In this setting, a classification of curves was speedily obtained, along with a good understanding of their singular points.

A substantial subsidiary theme is the study of complex algebraic surfaces. and past attempts (slow and often frustrated) to generalize the theory of curves.

The common thread is the Riemann -- Roch theorem. The author discusses the definition, proof, applications, interpretations and generalization of this theorem.

In addition, he gives an overview of the merits of the rival claims and approaches made by algebraists and geometers.

This important feature of mathematical debate has been widely observed, but there has never been a historical account of it until now.

This volume also provides an account of the remarkable breakthrough in the 1950s which led to a steadily more general view of the Riemann -- Roch theorem, and which culminated in the development of the Atiyah -- Grothendieck -- Hirzebruch -- Riemann -- Roch theorem.