David Mumford

Archive for Reprints, Notes, Talks, and Blog

Professor Emeritus
Brown and Harvard Universities
David_Mumford@brown.edu

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Work on Curves and Surfaces

Curves and surfaces were the bread and butter of algebraic geometry from the 19th until the late 20th century and extending all the Italian and French results to characteristic p was a challenge that Oscar Zariski set for his students. My real conversion to the Grothendieck's way of thinking was his purely algebraic and transparent proof of the central result in the Italian theory of surfaces -- the completeness of the characteristic linear system of a complete algebraic system on any surface. His approach also leads to necessary and sufficient conditions for the theorem to hold in characteristic p. This is described in the third item below.

  • Topology of Normal Singularities and a Criterion for Simplicity, Publ. de l'Institut des Hautes Etudes Scientifiques, 1961, pp. 5-22.Scanned reprint and DASH reprint
  • The Canonical Ring of an Algebraic Surface, an appendix to a paper by Oscar Zariski, Ann. of Math., 76, 1962, pp. 612-615.Scanned reprint
  • Lectures on Curves on Surfaces, with George Bergman, Princeton University Press, 1964.
  • Enriques' Classification of Surfaces in Char. p, I, in Global Analysis (papers in honor of K. Kodaira), Spencer and Iyanaga editors, U. of Tokyo Press, 1969, pp. 325-339; Enriques Classification of Surfaces in Char. p, II (with Enrico Bombieri), in Complex Analysis and Algebraic Geometry, Baily and Shioda editors, Cambridge Univ. Press, 1977, pp. 23-42; Enriques Classification of Surfaces in Char. p, III (with Enrico Bombieri), Inventiones Mathematicae, 1976, 35, pp. 197-232. Scanned reprint part I, Scanned reprint part II, Scanned reprint part III and DASH reprint (Part III only)
  • Rational Equivalences of 0-cycles on Surfaces, J. of Math. of Kyoto Univ., 1969, 9, pp. 195-204. Scanned reprint
  • Six Appendices to Algebraic Surfaces, by O. Zariski, 2nd edition, Springer-Verlag, 1971.
  • Theta Characteristics of an Algebraic Curve, Annales de l'Ecole Norm. Sup., 1971, pp. 181-192. Scanned reprint and DASH reprint
  • An Analytic Construction of Degenerating Curves Over Complete Local Rings, Composito Math., 1972, 24, pp. 129-174.Scanned reprint
  • Prym Varieties I, in Contributions to Analysis, Academic Press, 1974, pp. 325-350. Scanned reprint
  • Curves and their Jacobians, University of Michigan Press, 1975, reprinted as part of the 'Red Book', 1999.

These rice powder designs are made everywhere in India and have many names. Called a 'kolam' in Tamil, they are remade every morning by a housewife. Many different designs have been passed down from mother to daughter for centuries.

A plane quartic (the black curve) with its 28 bitangents

A plane quartic with its 28 bitangents, all real in this case.

Cover of the book

Lectures on Curves on Surfaces, with George Bergman.