DOCUMENTA MATHEMATICA, Extra Volume ICM III (1998), 811-822

Jeremy J Gray

Title: The Riemann-Roch Theorem and Geometry, 1854-1914

The history of the Riemann-Roch Theorem, from its discovery by Riemann and Roch in the 1850's to its use by Castelnuovo and Enriques in from 1890 to 1914, offers one of the most instructive examples in the history of mathematics of how a result stays alive in mathematics by admitting many interpretations. Various mathematicians over the years took the theorem to be central to their researches in complex function theory, and in the study of algebraic curves and surfaces in a variety of algebraic and geometric styles. In surveying their interpretations and extensions of the theorem, the historian traces the creation of a general theory of complex algebraic curves and surfaces in the period, and uncovers lively agreements and disagreements. This paper provides an overview of the field; the Congress lecture will concentrate on the route from Riemann and Roch via Brill and Noether to Castelnuovo and Enriques. For reasons of space a number of the better-known developments have been omitted. One may consult Dieudonn\'{e} [1976].

1991 Mathematics Subject Classification: 1, 14, 30

Keywords and Phrases: Riemann-Roch Theorem, algebraic curve, algebraic surface

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