History Algebraic Geometry

Suzanne C. Dieudonne, Jean Dieudonné

• Publisher: Chapman and Hall/CRC
• Year: 2017
• eBook ISBN: 9781351440530
• Topic: Mathematics
• Subtopic: Number Theory
• Index: Mathematics

This book contains several fundamental ideas that are revived time after time in different guises, providing a better understanding of algebraic geometric phenomena. It shows how the field is enriched with loans from analysis and topology and from commutative algebra and homological algebra.

IX —RECENT RESULTS AND OPEN PROBLEMS

1. Since 1960, there has been a tremendous increase in the number of mathe maticians who are working in algebraic geometry. A summer institute held in California in 1974 was attended by 270 mathematicians! This phenomenon can be attributed on one hand to the unusual wealth and diversity of (old and new) interesting and “natural” problems afforded by algebraic geometry, and to their multiple connections with number theory, analytic geometry, and analysis, and on the other hand to the availability of powerful and easy-to-grasp techniques of attack, with no need for an appeal to some personal and more or less reliable “intuition.”

It is of course utterly impossible to give an exhaustive treatment of all the results recently obtained in algebraic geometry. Here we can give only sketchy accounts of what seem to be the more significant developments, supplemented by bibliographical references to the original papers, or more often to expository books and papers, which themselves usually contain a much more extensive bibliography.

The division of this chapter into sections is purely for convenience, and their multiple interconnections will easily be perceived by the reader.

1. PROBLEMS ON CURVES

2. Many papers on algebraic geometry published since 1950 are concerned with problems on curves and surfaces in projective complex spaces P_n(C) that already had been studied by the Italian geometers. But many of the Italian proofs were inconclusive, or based on “intuitive” arguments, and it was necessary to put them on more secure foundations, chiefly based on sheaf cohomology and other modern techniques, such as the theory of abelian varieties. Another problem was to see if the results of the Italians generalized to projective curves or surfaces over an arbitrary algebraically closed field of characteristic p > 0.

3. A) Special Divisors. Let be a smooth projective complex curve of genus g ≥ 2. For any divisor D on C, let r(D) = l(D) — 1 be the dimension of the projective space |D| (VI, 9). Recall that if Δ is a canonical divisor on (VI, 13), the Riemann-Roch theorem (VI, 14) gives

(1)r(D)=d-g+l(Δ-D)

where d = deg D. The divisor D is called special if l(Δ — D) > 0. For d > 0, Clifford’s theorem (VI, 14) shows that

(2)r(D)≤d/2≤g-1.

One may show that on a hyperelliptic curve (VI, 15), for all pairs (r, d) such that r ≤ d/2 ≤ g – 1, there exist divisors D of degree d with r(D) = r. But what can be said for special divisors on a nonhyperelliptic curve (g ≥ 3) ; do they exist and “how many” of them are there? This problem occupied several mathematicians in the period 1960-1980, and it seems worthwhile to describe its solution in some detail, in order to exhibit the constant interplay between the geometric insights of the Italians and the modern tools of the theory of schemes that made them effective.

4. To put these questions in a precise form, we introduce the symmetric product C_d of d copies of (the variety of orbits of the symmetric group Sd acting naturally on the product C^d), which parametrizes the set of all divisors of degree d on C. Let Cdr be the subvariety of C_d consisting of the divisors D such that r(D) ≥ r, and let Wdf be the natural image of Cdr in the jacobian J(C) (variety of classes of divisors of Cdf for linear equivalence (VII, 57)). The precise form of the problem of special divisors is then to determine the dimensions...

Table of contents

• Cover Page
• Half Title
• Title Page
• Copyright Page
• Contents
• Foreword
• I. Introduction
• Table of Notations
• II. The First Epoch—Prehistory (ca. 400 B.c.-1630 A.D.)
• III. The Second Epoch—Exploration (1630-1795)
• IV. The Third Epoch—The golden age of projective geometry (1795-1850)
• V. The Fourth Epoch—Riemann and birational geometry (1850-1866)
• VI. The Fifth Epoch—Development and Chaos (1866-1920)
• VII. The Sixth Epoch—New Structures in Algebraic Geometry (1920-1950)
• VIII. The Seventh Epoch—Sheaves and Schemes (1950- )
• IX. Recent Results and Open Problems
• Annotated Bibliography
• Index of Cited Names
• Index of Terminology