|
Rational Map
In mathematics, in particular the subfield of algebraic geometry, a rational map or rational mapping is a kind of partial function between algebraic varieties. This article uses the convention that varieties are irreducible. Definition Formal definition Formally, a rational map f \colon V \to W between two varieties is an equivalence class of pairs (f_U, U) in which f_U is a morphism of varieties from a non-empty open set U\subset V to W, and two such pairs (f_U, U) and (_, U') are considered equivalent if f_U and _ coincide on the intersection U \cap U' (this is, in particular, vacuously true if the intersection is empty, but since V is assumed irreducible, this is impossible). The proof that this defines an equivalence relation relies on the following lemma: * If two morphisms of varieties are equal on some non-empty open set, then they are equal. f is said to be birational if there exists a rational map g \colon W \to V which is its inverse, where the composition is taken i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field Extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ''F''. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers. Field extensions are fundamental in algebraic number theory, and in the study of polynomial roots through Galois theory, and are widely used in algebraic geometry. Subfield A subfield K of a field L is a subset K\subseteq L that is a field with respect to the field operations inherited from L. Equivalently, a subfield is a subset that contains 1, and is closed under the operations of addition, subtraction, multiplication, and taking the inverse of a nonzero element of K. As , the latter definition implies K and L have the same zero eleme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry (book)
''Algebraic Geometry'' is an algebraic geometry textbook written by Robin Hartshorne and published by Springer-Verlag in 1977.MathSciNet lists more than 2500 citations of this book. Importance It was the first extended treatment of scheme theory written as a text intended to be accessible to graduate students. Contents The first chapter, titled "Varieties", deals with the classical algebraic geometry of varieties over algebraically closed fields. This chapter uses many classical results in commutative algebra, including Hilbert's Nullstellensatz In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic ..., with the books by Atiyah–Macdonald, Matsumura, and Zariski–Samuel as usual references. The second and the third chapters, "Schemes" and "Cohomology", form the technical heart of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Log Structure
In algebraic geometry, a log structure provides an abstract context to study semistable schemes, and in particular the notion of logarithmic form, logarithmic differential form and the related Hodge theory, Hodge-theoretic concepts. This idea has applications in the theory of moduli spaces, in deformation theory and Fontaine's p-adic Hodge theory, among others. Motivation The idea is to study some algebraic variety (or scheme (mathematics), scheme) ''U'' which is smooth morphism, smooth but not necessarily proper morphism, proper by embedding it into ''X'', which is proper, and then looking at certain sheaves on ''X''. The problem is that the subsheaf of \mathcal_X consisting of functions whose restriction to ''U'' is invertible is not a sheaf of rings (as adding two non-vanishing functions could provide one which vanishes), and we only get a sheaf of submonoids of \mathcal_X , multiplicatively. Remembering this additional structure on ''X'' corresponds to remembering the inclu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. Outline 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". The precise meaning of this phrase has evolved with the development of the subject; originally for surfaces, it meant finding a smooth variety X for which any birational morphism f\colon X \to X' with a smooth surface X' is an isomorphism. In the modern formulation, the goal of the theory is as follows. Suppose we are given a projective variety X, which for simplicity is assumed non-singular. There are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. Definition for complex manifolds 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. The function field of a variety is then the set of all meromorphic functions on the variety. (Like all meromorphic functions, these take their values in \mathbb\cup\infty.) Together with the operations of addition and multiplication of functions, this is a field in the sense of algebra. For the Riemann sphere, which is the variety \mathbb^1 over the complex numbers, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Blowing Up
In mathematics, blowing up or blowup is a type of geometric transformation which replaces a subspace of a given space with all the directions pointing out of that subspace. For example, the blowup of a point in a plane replaces the point with the projectivized tangent space at that point. The metaphor is that of zooming in on a photograph to enlarge part of the picture, rather than referring to an explosion. Blowups are the most fundamental transformation in birational geometry, because every birational morphism between projective varieties is a blowup. The weak factorization theorem says that every birational map can be factored as a composition of particularly simple blowups. The Cremona group, the group of birational automorphisms of the plane, is generated by blowups. Besides their importance in describing birational transformations, blowups are also an important way of constructing new spaces. For instance, most procedures for resolution of singularities proceed by bl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birational Geometry
In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational functions rather than polynomials; the map may fail to be defined where the rational functions have poles. Birational maps Rational maps A rational map from one variety (understood to be irreducible) X to another variety Y, written as a dashed arrow , is defined as a morphism from a nonempty open subset U \subset X to Y. By definition of the Zariski topology used in algebraic geometry, a nonempty open subset U is always dense in X, in fact the complement of a lower-dimensional subset. Concretely, a rational map can be written in coordinates using rational functions. Birational maps A birational map from ''X'' to ''Y'' is a rational map such that there is a rational map inverse to ''f''. A birational map induces an isomorphism from a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field Of Fractions
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field of rational numbers. Intuitively, it consists of ratios between integral domain elements. The field of fractions of R is sometimes denoted by \operatorname(R) or \operatorname(R), and the construction is sometimes also called the fraction field, field of quotients, or quotient field of R. All four are in common usage, but are not to be confused with the quotient of a ring by an ideal, which is a quite different concept. For a commutative ring which is not an integral domain, the analogous construction is called the localization or ring of quotients. Definition Given an integral domain and letting R^* = R \setminus \, we define an equivalence relation on R \times R^* by letting (n,d) \sim (m,b) whenever nb = md. We denote the equivale ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 exists a non-negative integer ''d'' and algebraically independent elements ''y''1, ''y''2, ..., ''y''''d'' in ''A'' such that ''A'' is a finitely generated module over the polynomial ring ''S'' = ''k'' 'y''1, ''y''2, ..., ''y''''d'' The integer ''d'' above is uniquely determined; it is the Krull dimension of the ring ''A''. When ''A'' is an integral domain, ''d'' is also the transcendence degree of the field of fractions of ''A'' over ''k''. The theorem has a geometric interpretation. Suppose ''A'' is integral. Let ''S'' be the coordinate ring of the ''d''-dimensional affine space \mathbb A^d_k, and let ''A'' be the coordinate ring of some other ''d''-dimensional affine variety ''X''. Then the inclusion map ''S'' → ''A'' induces a surjective f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Resolution Of Singularities
In algebraic geometry, the problem of resolution of singularities asks whether every algebraic variety ''V'' has a resolution, a non-singular variety ''W'' with a proper birational map ''W''→''V''. For varieties over fields of characteristic 0 this was proved in Hironaka (1964), while for varieties over fields of characteristic ''p'' it is an open problem in dimensions at least 4. Definitions Originally the problem of resolution of singularities was to find a nonsingular model for the function field of a variety ''X'', in other words a complete non-singular variety ''X′'' with the same function field. In practice it is more convenient to ask for a different condition as follows: a variety ''X'' has a resolution of singularities if we can find a non-singular variety ''X′'' and a proper birational map from ''X′'' to ''X''. The condition that the map is proper is needed to exclude trivial solutions, such as taking ''X′'' to be the subvariety of non- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cubic Surface
In mathematics, a cubic surface is a surface in 3-dimensional space defined by one polynomial equation of degree 3. Cubic surfaces are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than affine space, and so cubic surfaces are generally considered in projective 3-space \mathbf^3. The theory also becomes more uniform by focusing on surfaces over the complex numbers rather than the real numbers; note that a complex surface has real dimension 4. A simple example is the Fermat cubic surface :x^3+y^3+z^3+w^3=0 in \mathbf^3. Many properties of cubic surfaces hold more generally for del Pezzo surfaces. Rationality of cubic surfaces A central feature of smooth cubic surfaces ''X'' over an algebraically closed field is that they are all rational, as shown by Alfred Clebsch in 1866. That is, there is a one-to-one correspondence defined by rational functions between the projective plane \mathbf^2 minus a lower-dimensional subset and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
