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Über dieses Buch

This book is based on one-semester courses given at Harvard in 1984, at Brown in 1985, and at Harvard in 1988. It is intended to be, as the title suggests, a first introduction to the subject. Even so, a few words are in order about the purposes of the book. Algebraic geometry has developed tremendously over the last century. During the 19th century, the subject was practiced on a relatively concrete, down-to-earth level; the main objects of study were projective varieties, and the techniques for the most part were grounded in geometric constructions. This approach flourished during the middle of the century and reached its culmination in the work of the Italian school around the end of the 19th and the beginning of the 20th centuries. Ultimately, the subject was pushed beyond the limits of its foundations: by the end of its period the Italian school had progressed to the point where the language and techniques of the subject could no longer serve to express or carry out the ideas of its best practitioners.

Inhaltsverzeichnis

Frontmatter

Examples of Varieties and Maps

Frontmatter

Lecture 1. Affine and Projective Varieties

Abstract
In this book we will be dealing with varieties over a field K, which we will take to be algebraically closed throughout. Algebraic geometry can certainly be done over arbitrary fields (or even more generally over rings), but not in so straightforward a fashion as we will do here; indeed, to work with varieties over nonalgebraically closed fields the best language to use is that of scheme theory. Classically, much of algebraic geometry was done over the complex numbers ℂ, and this remains the source of much of our geometric intuition; but where possible we will avoid assuming K = ℂ.
Joe Harris

Lecture 2. Regular Functions and Maps

Abstract
In the preceding lecture, we introduced the basic objects of the category we will be studying; we will now introduce the maps. As might be expected, this is extremely easy in the context of affine varieties and slightly trickier, at least at first, for projective ones.
Joe Harris

Lecture 3. Cones, Projections, and More About Products

Abstract
We start here with a hyperplane ℙ n-1 ⊂ ℙ n and a point p ∈ ℙ n not lying on ℙ n-1; if we like, we can take coordinates Z on ℙ n so that ℙ n-1 is given by Z n = 0 and the point p= [0,..., 0, 1]. Let X ⊂ ℙ n-1 be any variety.
Joe Harris

Lecture 4. Families and Parameter Spaces

Abstract
Next, we will give a definition without much apparent content, but one that is fundamental in much of algebraic geometry. Basically, the situation is that, given a collection {V b } of projective varieties V b ⊂ ℙ n indexed by the points b of a variety B, we want to say what it means for the collection {V b} to “vary algebraically with parameters.” The answer is simple: for any variety B, we define a family of projective varieties in ℙ n with base B to be simply a closed subvariety V of the product B × ℙ n . The fibers V b = (π1)-1(b) of V over points of b are then referred to as the members, or elements of the family; the variety V is called the total space, and the family is said to be parametrized by B. The idea is that if B ⊂ ℙ m is projective, the family V m × ℙ n will be described by a collection of polynomials F α (Z, W) bihomogeneous in the coordinates Z on ℙ m and W on ℙ n , which we may then think of as a collection of polynomials in W whose coefficients are polynomials on B; similarly, if B is affine we may describe V by a collection of polynomials F α (z, W), which we may think of as homogeneous polynomials in the variables W whose coefficients are regular functions on B.
Joe Harris

Lecture 5. Ideals of Varieties, Irreducible Decomposition, and the Nullstellensatz

Abstract
The time has come to talk about the various senses in which a variety may be defined by a set of equations. There are three different meanings of the statement that a collection of polynomials {F α (Z)} “cut out” a variety X ⊂ ℙ n , and several different terms are used to convey each of these meanings.
Joe Harris

Lecture 6. Grassmannians and Related Varieties

Abstract
Grassmannians are fundamental objects in algebraic geometry: they are simultaneously objects of interest in their own right and basic tools in the construction and study of other varieties. We will be dealing with Grassmannians constantly in the course of this book; here we introduce them and mention a few of their basic properties.
Joe Harris

Lecture 7. Rational Functions and Rational Maps

Abstract
Let X ⊂ A n be an irreducible affine variety. Since its coordinate ring A(X) is an integral domain, we can form its quotient field; this is called the rational function field of X and is usually denoted K(X); its elements are called rational functions on X. Note that if Y ⊂ X is an open subset that is an affine variety in its own right (as in the discussion on page 19), the function field of Y will be the same as that of X.
Joe Harris

Lecture 8. More Examples

Abstract
Now that we have developed a body of fundamental notions, we are able to make a number of standard constructions.
Joe Harris

Lecture 9. Determinantal Varieties

Abstract
In this lecture we will introduce a large and important class of varieties, those whose equations take the form of the minors of a matrix. We will see that many of the varieties we have looked at so far—Veronese varieties, Segre varieties, rational normal scrolls, for example — are determinantal.
Joe Harris

Lecture 10. Algebraic Groups

Abstract
As with most geometric categories, we have the notion of a group object in classical algebraic geometry.
Joe Harris

Attributes of Varieties

Frontmatter

Lecture 11. Definitions of Dimension and Elementary Examples

Abstract
We will start by giving a number of different definitions of dimension and we will try to indicate how they relate to one another. All of our definitions initially apply to an irreducible variety X; the dimension of an arbitrary variety will be defined to be the maximum of the dimensions of its irreducible components.
Joe Harris

Lecture 12. More Dimension Computations

Abstract
As in Lecture 9, let M be the projective space of nonzero m × n matrices up to scalars and M k ⊂ M the variety of matrices of rank k or less.
Joe Harris

Lecture 13. Hilbert Polynomials

Abstract
Given that a projective variety X ⊂ ℙ n is an intersection of hypersurfaces, one of the most basic problems we can pose in relation to X is to describe the hypersurfaces that contain it. In particular, we want to know how many hypersurfaces of each degree contain X—that is, for each value of m, to know the dimension of the vector space of homogeneous polynomials of degree m vanishing on X.
Joe Harris

Lecture 14. Smoothness and Tangent Spaces

Abstract
The basic definition of a smooth point of an algebraic variety is analogous to the corresponding one from differential geometry. We start with the affine case; suppose X ⊂A n is an affine variety of pure dimension k, with ideal I(X) =(f 1 ,..., f i ). Let M be the l × n matrix with entries ∂f i /∂x j . Then it’s not hard to see that the rank of M is at most nk at every point of X, and we say a point pX is a smooth point of X if the rank of the matrix M, evaluated at the point p, exactly nk. Note that in case the ground field K = ℂ, this is equivalent to saying that X is a complex submanifold of An = ℂn in a neighborhood of p, or that X is a real submanifold of ℂn near p. (It is not, however, equivalent, in the case of a variety X defined by polynomials f α with real coefficients, to saying that the locus of the f α in ℝ n is smooth; consider, for example, the origin p = (0, 0) on the plane curve x 3 + y 3 = 0.)
Joe Harris

Lecture 15. Gauss Maps, Tangential and Dual Varieties

Abstract
In the preceding lecture, we associated to each point of a projective variety X ⊂ ℙ n a linear subspace of ℙ n. We investigate here how those planes vary on X, that is, the geometry of the Gauss map. Before we launch into this, however, we should take a moment to discuss a question that will be increasingly relevant to our analysis; the choice of our ground field K and in particular its characteristic.
Joe Harris

Lecture 16. Tangent Spaces to Grassmannians

Abstract
We have seen that the Grassmannian 𝔾(k, n) is a smooth variety of dimension (k + 1) (n - k). This follows initially from our explicit description of the covering of 𝔾 (k, n) by open sets U Λ ≅ 𝔸(k+1)(n-k), though we could also deduce this from the fact that it is a homogeneous space for the algebraic group PGL n+1 K. The Zariski tangent spaces to G are thus all vector spaces of this dimension. For many reasons, however, it is important to have a more intrinsic description of the space T Λ(𝔾;) in terms of the linear algebra of Λ ⊂ K n+1. We will derive such an expression here and then use it to describe the tangent spaces of the various varieties constructed in Part I with the use of the Grassmannians.
Joe Harris

Lecture 17. Further Topics Involving Smoothness and Tangent Spaces

Without Abstract
Joe Harris

Lecture 18. Degree

Abstract
Our next fundamental notion, that of the degree of a projective variety X ⊂ ℙ n , may be defined in ways for the most part exactly analogous to the notion of dimension. As in the case of the various definitions of dimension, the fact that these definitions are equivalent (or, in some cases, that they are well-defined at all) will not be established until we have introduced them all.
Joe Harris

Lecture 19. Further Examples and Applications of Degree

Abstract
Since finding out that a quadric surface Q ⊂3 is abstractly isomorphic to the product ℙ1 × ℙ1 we have observed a number of times that, in describing a curve C ⊂ Q, it is much more useful to give its bidegree (a, b) in ℙ1 × ℙ1 (that is, the bidegree of the bihomogeneous polynomial F defining it as a subvariety of ℙ1 ⊂ ℙ1) than to give just its degree as a curve in P3 (the reader can check this is just a + b). We ask now what are the analogous numerical invariants of a k-dimensional subvariety X ⊂ ℙ m × ℙ n in general.
Joe Harris

Lecture 20. Singular Points and Tangent Cones

Abstract
The Zariski tangent space to a variety X ⊂ 𝔸n at a point p is described by taking the linear part of the expansion around p of all the functions on 𝔸n vanishing on X. In case p is a singular point of X, however, this does not give us a very refined picture of the local geometry of X; for example, if X ⊂ 𝔸2 is a plane curve, the Zariski tangent space to X at any singular point p will be all of T p (𝔸 2) = K 2. We will describe here the tangent cone, an object that, while it certainly does not give a complete description of the local structure of a variety at a singular point, is at least a partial refinement of the notion of tangent space.
Joe Harris

Lecture 21. Parameter Spaces and Moduli Spaces

Abstract
We can now give a slightly expanded introduction to the notion of parameter space, introduced in Lecture 4 and discussed occasionally since. This is a fairly delicate subject, and one that is clearly best understood from the point of view of scheme theory, so that in some sense this discussion violates our basic principle of dealing only with topics that can be reasonably well understood on an elementary level. Nevertheless, since it is one of the fundamental constructions of algebraic geometry, and since the constructions can at least be described in an elementary fashion, we will proceed. One unfortunate consequence of this sort of overreaching, however, is that the density of unproved assertions, high enough in the rest of the text, will reach truly appalling levels in this lecture.
Joe Harris

Lecture 22. Quadrics

Abstract
In this final lecture, we will study in more detail what are perhaps the simplest and most fundamental of all varieties: quadric hypersurfaces. The idea is partly to become familiar with these basic objects and partly to see some of the ideas we have studied in the preceding lectures applied. In the course of this (somewhat lengthy) lecture, we will involve the notions of dimensions, degree, rational maps, smoothness and singularity, tangent spaces and tangent cones, Fano varieties, and families —all in the context of the analysis of one class of objects. This is a much less technically demanding lecture than the last; we are not pushing the boundaries of what we can do with available techniques here, but carrying out a classical and elementary investigation.
Joe Harris

Backmatter

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