Concept
Morphism of algebraic varieties
Related concepts (24)
Noether normalization lemma
In mathematics, the Noether normalization lemma is a result of commutative algebra, introduced by Emmy Noether in 1926. It states that for any field k, and any finitely generated commutative k-algebra A, there exist algebraically independent elements y1, y2, ..., yd in A such that A is a finitely generated module over the polynomial ring S = k [y1, y2, ..., yd]. The integer d is equal to the Krull dimension of the ring A; and if A is an integral domain, d is also the transcendence degree of the field of fractions of A over k.
Function field of an algebraic variety
In algebraic geometry, the function field of an algebraic variety V consists of objects which are interpreted as rational functions on V. In classical algebraic geometry they are ratios of polynomials; in complex algebraic geometry these are meromorphic functions and their higher-dimensional analogues; in modern algebraic geometry they are elements of some quotient ring's field of fractions. In complex algebraic geometry the objects of study are complex analytic varieties, on which we have a local notion of complex analysis, through which we may define meromorphic functions.
Minimal model program
In algebraic geometry, the minimal model program is part of the birational classification of algebraic varieties. Its goal is to construct a birational model of any complex projective variety which is as simple as possible. The subject has its origins in the classical birational geometry of surfaces studied by the Italian school, and is currently an active research area within algebraic geometry. The basic idea of the theory is to simplify the birational classification of varieties by finding, in each birational equivalence class, a variety which is "as simple as possible".
Normal scheme
In algebraic geometry, an algebraic variety or scheme X is normal if it is normal at every point, meaning that the local ring at the point is an integrally closed domain. An affine variety X (understood to be irreducible) is normal if and only if the ring O(X) of regular functions on X is an integrally closed domain. A variety X over a field is normal if and only if every finite birational morphism from any variety Y to X is an isomorphism. Normal varieties were introduced by .
Rational variety
In mathematics, a rational variety is an algebraic variety, over a given field K, which is birationally equivalent to a projective space of some dimension over K. This means that its function field is isomorphic to the field of all rational functions for some set of indeterminates, where d is the dimension of the variety. Let V be an affine algebraic variety of dimension d defined by a prime ideal I = ⟨f1, ..., fk⟩ in . If V is rational, then there are n + 1 polynomials g0, ..., gn in such that In order words, we have a of the variety.
Affine variety
In algebraic geometry, an affine algebraic set is the set of the common zeros over an algebraically closed field k of some family of polynomials in the polynomial ring An affine variety or affine algebraic variety, is an affine algebraic set such that the ideal generated by the defining polynomials is prime. Some texts call variety any algebraic set, and irreducible variety an algebraic set whose defining ideal is prime (affine variety in the above sense).
Étale morphism
In algebraic geometry, an étale morphism (etal) is a morphism of schemes that is formally étale and locally of finite presentation. This is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy the hypotheses of the implicit function theorem, but because open sets in the Zariski topology are so large, they are not necessarily local isomorphisms. Despite this, étale maps retain many of the properties of local analytic isomorphisms, and are useful in defining the algebraic fundamental group and the étale topology.
Smooth morphism
In algebraic geometry, a morphism between schemes is said to be smooth if (i) it is locally of finite presentation (ii) it is flat, and (iii) for every geometric point the fiber is regular. (iii) means that each geometric fiber of f is a nonsingular variety (if it is separated). Thus, intuitively speaking, a smooth morphism gives a flat family of nonsingular varieties. If S is the spectrum of an algebraically closed field and f is of finite type, then one recovers the definition of a nonsingular variety.
Algebraic function field
In mathematics, an algebraic function field (often abbreviated as function field) of n variables over a field k is a finitely generated field extension K/k which has transcendence degree n over k. Equivalently, an algebraic function field of n variables over k may be defined as a finite field extension of the field K = k(x1,...,xn) of rational functions in n variables over k. As an example, in the polynomial ring k [X,Y] consider the ideal generated by the irreducible polynomial Y 2 − X 3 and form the field of fractions of the quotient ring k [X,Y]/(Y 2 − X 3).
Smooth scheme
In algebraic geometry, a smooth scheme over a field is a scheme which is well approximated by affine space near any point. Smoothness is one way of making precise the notion of a scheme with no singular points. A special case is the notion of a smooth variety over a field. Smooth schemes play the role in algebraic geometry of manifolds in topology. First, let X be an affine scheme of finite type over a field k. Equivalently, X has a closed immersion into affine space An over k for some natural number n.