Algebraic Geometry

An Introduction

verfasst von: Daniel Perrin

Verlag: Springer London

Buchreihe : Universitext

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Über dieses Buch

This book is built upon a basic second-year masters course given in 1991– 1992, 1992–1993 and 1993–1994 at the Universit´ e Paris-Sud (Orsay). The course consisted of about 50 hours of classroom time, of which three-quarters were lectures and one-quarter examples classes. It was aimed at students who had no previous experience with algebraic geometry. Of course, in the time available, it was impossible to cover more than a small part of this ?eld. I chose to focus on projective algebraic geometry over an algebraically closed base ?eld, using algebraic methods only. The basic principles of this course were as follows: 1) Start with easily formulated problems with non-trivial solutions (such as B´ ezout’s theorem on intersections of plane curves and the problem of rationalcurves).In1993–1994,thechapteronrationalcurveswasreplaced by the chapter on space curves. 2) Use these problems to introduce the fundamental tools of algebraic ge- etry: dimension, singularities, sheaves, varieties and cohomology. I chose not to explain the scheme-theoretic method other than for ?nite schemes (which are needed to be able to talk about intersection multiplicities). A short summary is given in an appendix, in which special importance is given to the presence of nilpotent elements. 3) Use as little commutative algebra as possible by quoting without proof (or proving only in special cases) a certain number of theorems whose proof is not necessary in practise. The main theorems used are collected in a summary of results from algebra with references. Some of them are suggested as exercises or problems.

Inhaltsverzeichnis

Frontmatter

1. Affine algebraic sets

Let n be a positive integer. Consider the space¹kⁿ. If x = (x₁, …, x_n) is a point in kⁿ and P(X₁, …, X_n) is a polynomial, we denote P(x₁, …, x_n) by P(x). The first fundamental objects we encounter are the affine algebraic sets defined below.

2. Projective algebraic sets

We have already seen the main reason for introducing projective space in the Introduction when discussing Bézout's theorem. In affine space, results on intersections always contain a certain number of special cases due to parallel lines or asymptotes. For example, in the plane two distinct lines meet at a unique point except when they are parallel. In projective space, there are no such exceptions.

3. Sheaves and varieties

If we compare the study of affine algebraic sets and projective algebraic sets, we find many similarities and a few fundamental differences, such as the role played by homogeneous polynomials and graded rings in projective geometry. The most important difference, however, is the functions. If V is an affine algebraic set, we have a lovely function algebra Γ(V) and an almost perfect dictionary translating properties of V into properties of Γ(V). One of the problems of projective geometry is that elements of Γ_h(V) do not define functions on V, even in the simplest case, namely a homogeneous polynomial, since if x ∈ Pⁿ and F is homogeneous of degree d, then the quantity F(x) depends on the choice of representative: F(λx) = λ^dF(x).

4. Dimension

Dimension is the first and most natural invariant of an algebraic variety. We will finally be able to talk about varieties of dimension 0 (points), 1 (curves) and 2 (surfaces)… We will give a very natural topological definition of dimension, which is not always easy to work with, followed by other definitions which are easier to work with but which depend on results from algebra.

5. Tangent spaces and singular points

We start with a little differential geometry. Let f(x₁, …, x_n) = 0 be a hypersurface S ⊂ Rⁿ. We assume that f is C∞. Consider a = (a₁,…,a_n) ∈ S. What is the tangent space to S at a?

6. Bézout's theorem

Our aim is to show that two plane curves of degrees s and t have exactly st intersection points. We saw in the introduction that we need to take care when stating this result.

7. Sheaf cohomology

We return for a moment to the proof of Bézout's theorem. Given Z = V (F,G) we had to calculate the dimension of Γ(Z,O_Z). The method used was to consider the exact sequences

8. Arithmetic genus of curves and the weak Riemann-Roch theorem

If F is a coherent sheaf over a projective variety X, then we have seen that HⁱF = Hⁱ(X,F) is a finite-dimensional k-vector space. Our aim is to calculate its dimension hⁱF. We will see below that this situation often arises in practice. This is not an easy problem in general. There is, however, an invariant of F which is much easier to calculate than the numbers hⁱF, namely the Euler-Poincaré characteristic X(F) = Σ_i⩾0 (−1)ⁱhⁱF.

9. Rational maps, geometric genus and rational curves

We saw in the book's introduction how useful it can be to have rational parameterisations of curves (notably for resolving Diophantine equations or calculating primitives). We then say the curve is rational. The aim of this chapter is to give a method for calculating whether or not a curve is rational. We will prove that this is equivalent to the (geometric) genus of the curve being zero and we will give methods for calculating this geometric genus.

10. Liaison of space curves

We assume that the field k is algebraically closed. We will use the following notation:

Backmatter

Titel: Algebraic Geometry
verfasst von: Daniel Perrin
Verlag: Springer London
Electronic ISBN: 978-1-84800-056-8
Print ISBN: 978-1-84800-055-1
DOI: https://doi.org/10.1007/978-1-84800-056-8