Algebraic Geometry

Solomon Lefschetz

• Publisher: Dover Publications
• Year: 2012
• ISBN:
  9780486154725
• Topic: Matematica
• Subtopic: Geometria algebrica

This text for advanced undergraduate students is both an introduction to algebraic geometry and a bridge between its two parts — the analytical-topological and the algebraic. Because of its extensive use of formal power series (power series without convergency), the treatment will appeal to readers conversant with analysis but less familiar with the formidable techniques of modern algebra.
The book opens with an overview of the results required from algebra and proceeds to the fundamental concepts of the general theory of algebraic varieties: general point, dimension, function field, rational transformations, and correspondences. A concentrated chapter on formal power series with applications to algebraic varieties follows. An extensive survey of algebraic curves includes places, linear series, abelian differentials, and algebraic correspondences. The text concludes with an examination of systems of curves on a surface.

I. Algebraic Foundations

§1. PRELIMINARIES

1. The reader is expected to be familiar with the elementary concepts of modern algebra: groups, rings, ideals, fields, and likewise with the customary notations of the subject. Multiplication is supposed to be commutative throughout. To avoid certain awkward points appeal is made to the well known device of an all embracing field Ω which includes all the elements of rings,· · ·, under consideration.

(1.1) Notations. Aggregates such as x₀, · · · , x_n or α₁, · · · , α_m will often be written x or α, the range being generally clear from the context. Accordingly the ring or field extensions K[x₀, · · · , x_n] or K(α₁ · · · , α_m) will be written K[x] or K(α), with evident variants of these designations. Similarly for example for the functional notations: f(x) or φ(α.) for f(x₀, · · ·, x_n), or φ(α₁ · · · , α_m). In this connection the “partial” extensions K[x₀, · · · , x_r], K(α₁, · · · , α_s), will also be written K^r[x], K^s(α), with meaning generally clear from the context.

The following symbols of point-set theory will also be utilized throughout:

⊂: is contained in; ⊃: contains;

: intersection,

union; ∈ : is an element of.

(1.2) The groundfield. Very soon a certain fundamental field K, the groundfield will dominate the situation and all rings and fields will then be extensions of K. When K is of characteristic p the universal field Ω is also supposed to be of the same characteristic. The groundfield is always assumed to be infinite and perfect (irreducible polynomials have no multiple roots in an algebraic extension of K). Often also K is supposed to be algebraically closed (polynomials with coefficients in K have all their...