In classical algebraic geometry, the main objects of interest are the vanishing sets of collections of polynomials , meaning the set of all points that simultaneously satisfy one or more polynomial equations. For instance, the two-dimensional sphere of radius 1 in three-dimensional Euclidean space R 3 could be defined as the set of all points x , y , z with.
A "slanted" circle in R 3 can be defined as the set of all points x , y , z which satisfy the two polynomial equations. First we start with a field k. In classical algebraic geometry, this field was always the complex numbers C , but many of the same results are true if we assume only that k is algebraically closed. We consider the affine space of dimension n over k , denoted A n k or more simply A n , when k is clear from the context.
When one fixes a coordinate system, one may identify A n k with k n. The purpose of not working with k n is to emphasize that one "forgets" the vector space structure that k n carries. The property of a function to be polynomial or regular does not depend on the choice of a coordinate system in A n. When a coordinate system is chosen, the regular functions on the affine n -space may be identified with the ring of polynomial functions in n variables over k.
Therefore, the set of the regular functions on A n is a ring, which is denoted k [ A n ]. We say that a polynomial vanishes at a point if evaluating it at that point gives zero. Let S be a set of polynomials in k [ A n ]. The vanishing set of S or vanishing locus or zero set is the set V S of all points in A n where every polynomial in S vanishes.
A subset of A n which is V S , for some S , is called an algebraic set. The V stands for variety a specific type of algebraic set to be defined below. Given a subset U of A n , can one recover the set of polynomials which generate it?
If U is any subset of A n , define I U to be the set of all polynomials whose vanishing set contains U. The I stands for ideal: The answer to the first question is provided by introducing the Zariski topology , a topology on A n whose closed sets are the algebraic sets, and which directly reflects the algebraic structure of k [ A n ]. The answer to the second question is given by Hilbert's Nullstellensatz. In one of its forms, it says that I V S is the radical of the ideal generated by S.
In more abstract language, there is a Galois connection , giving rise to two closure operators ; they can be identified, and naturally play a basic role in the theory; the example is elaborated at Galois connection. For various reasons we may not always want to work with the entire ideal corresponding to an algebraic set U.
Hilbert's basis theorem implies that ideals in k [ A n ] are always finitely generated. An algebraic set is called irreducible if it cannot be written as the union of two smaller algebraic sets.
Any algebraic set is a finite union of irreducible algebraic sets and this decomposition is unique. Thus its elements are called the irreducible components of the algebraic set. An irreducible algebraic set is also called a variety. It turns out that an algebraic set is a variety if and only if it may be defined as the vanishing set of a prime ideal of the polynomial ring. Some authors do not make a clear distinction between algebraic sets and varieties and use irreducible variety to make the distinction when needed.
Just as continuous functions are the natural maps on topological spaces and smooth functions are the natural maps on differentiable manifolds , there is a natural class of functions on an algebraic set, called regular functions or polynomial functions. A regular function on an algebraic set V contained in A n is the restriction to V of a regular function on A n.
For an algebraic set defined on the field of the complex numbers, the regular functions are smooth and even analytic. It may seem unnaturally restrictive to require that a regular function always extend to the ambient space, but it is very similar to the situation in a normal topological space , where the Tietze extension theorem guarantees that a continuous function on a closed subset always extends to the ambient topological space.
Just as with the regular functions on affine space, the regular functions on V form a ring, which we denote by k [ V ]. This ring is called the coordinate ring of V. Since regular functions on V come from regular functions on A n , there is a relationship between the coordinate rings. Using regular functions from an affine variety to A 1 , we can define regular maps from one affine variety to another.
First we will define a regular map from a variety into affine space: Let V be a variety contained in A n. Choose m regular functions on V , and call them f 1 , In other words, each f i determines one coordinate of the range of f. The definition of the regular maps apply also to algebraic sets. The regular maps are also called morphisms , as they make the collection of all affine algebraic sets into a category , where the objects are the affine algebraic sets and the morphisms are the regular maps.
The affine varieties is a subcategory of the category of the algebraic sets. This defines an equivalence of categories between the category of algebraic sets and the opposite category of the finitely generated reduced k -algebras.
This equivalence is one of the starting points of scheme theory. In contrast to the preceding sections, this section concerns only varieties and not algebraic sets. On the other hand, the definitions extend naturally to projective varieties next section , as an affine variety and its projective completion have the same field of functions. If V is an affine variety, its coordinate ring is an integral domain and has thus a field of fractions which is denoted k V and called the field of the rational functions on V or, shortly, the function field of V.
Its elements are the restrictions to V of the rational functions over the affine space containing V. The domain of a rational function f is not V but the complement of the subvariety a hypersurface where the denominator of f vanishes. As with regular maps, one may define a rational map from a variety V to a variety V '.
As with the regular maps, the rational maps from V to V ' may be identified to the field homomorphisms from k V ' to k V. Two affine varieties are birationally equivalent if there are two rational functions between them which are inverse one to the other in the regions where both are defined.
Equivalently, they are birationally equivalent if their function fields are isomorphic. An affine variety is a rational variety if it is birationally equivalent to an affine space.
This means that the variety admits a rational parameterization. The problem of resolution of singularities is to know if every algebraic variety is birationally equivalent to a variety whose projective completion is nonsingular see also smooth completion. It was solved in the affirmative in characteristic 0 by Heisuke Hironaka in and is yet unsolved in finite characteristic.
Just as the formulas for the roots of second, third, and fourth degree polynomials suggest extending real numbers to the more algebraically complete setting of the complex numbers, many properties of algebraic varieties suggest extending affine space to a more geometrically complete projective space. If we draw it, we get a parabola. As x goes to negative infinity, the slope of the same line goes to negative infinity.
This is a cubic curve. But unlike before, as x goes to negative infinity, the slope of the same line goes to positive infinity as well; the exact opposite of the parabola. The consideration of the projective completion of the two curves, which is their prolongation "at infinity" in the projective plane , allows us to quantify this difference: Also, both curves are rational, as they are parameterized by x , and the Riemann-Roch theorem implies that the cubic curve must have a singularity, which must be at infinity, as all its points in the affine space are regular.
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