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Rational Parametrization
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 :K(U_1, \dots , U_d), the field of all rational functions for some set \ of indeterminates, where ''d'' is the dimension of the variety. Rationality and parameterization Let ''V'' be an affine algebraic variety of dimension ''d'' defined by a prime ideal ''I'' = ⟨''f''1, ..., ''f''''k''⟩ in K _1, \dots , X_n/math>. If ''V'' is rational, then there are ''n'' + 1 polynomials ''g''0, ..., ''g''''n'' in K(U_1, \dots , U_d) such that f_i(g_1/g_0, \ldots, g_n/g_0)=0. In other words, we have a x_i=\frac(u_1,\ldots,u_d) of the variety. Conversely, such a rational parameterization induces a field homomorphism of the field of functions of ''V'' into K(U_1, \dots , U_d). But this homomorphism is not necessarily onto. If such a parameterizatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Function Field
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers; they may be taken in any field . In this case, one speaks of a rational function and a rational fraction ''over ''. The values of the variables may be taken in any field containing . Then the domain of the function is the set of the values of the variables for which the denominator is not zero, and the codomain is . The set of rational functions over a field is a field, the field of fractions of the ring of the polynomial functions over . Definitions A function f is called a rational function if it can be written in the form : f(x) = \frac where P and Q are polynomial functions of x and Q is not the zero function. The domain of f is the set of all values of x for which the denominator Q(x) is not zero. However, if ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plurigenus
In mathematics, the pluricanonical ring of an algebraic variety ''V'' (which is nonsingular), or of a complex manifold, is the graded ring :R(V,K)=R(V,K_V) \, of sections of powers of the canonical bundle ''K''. Its ''n''th graded component (for n\geq 0) is: :R_n := H^0(V, K^n),\ that is, the space of sections of the ''n''-th tensor product ''K''''n'' of the canonical bundle ''K''. The 0th graded component R_0 is sections of the trivial bundle, and is one-dimensional as ''V'' is projective. The projective variety defined by this graded ring is called the canonical model of ''V'', and the dimension of the canonical model is called the Kodaira dimension of ''V''. One can define an analogous ring for any line bundle In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organis ... ''L'' over ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Genus
In mathematics, the arithmetic genus of an algebraic variety is one of a few possible generalizations of the genus of an algebraic curve or Riemann surface. Projective varieties Let ''X'' be a projective scheme of dimension ''r'' over a field ''k'', the ''arithmetic genus'' p_a of ''X'' is defined asp_a(X)=(-1)^r (\chi(\mathcal_X)-1).Here \chi(\mathcal_X) is the Euler characteristic of the structure sheaf \mathcal_X. Complex projective manifolds The arithmetic genus of a complex projective manifold of dimension ''n'' can be defined as a combination of Hodge numbers, namely :p_a=\sum_^ (-1)^j h^. When ''n=1'', the formula becomes p_a=h^. According to the Hodge theorem, h^=h^. Consequently h^=h^1(X)/2=g, where ''g'' is the usual (topological) meaning of genus of a surface, so the definitions are compatible. When ''X'' is a compact Kähler manifold, applying ''h''''p'',''q'' = ''h''''q'',''p'' recovers the earlier definition for projective varieties. Kähler manifolds By u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann–Hurwitz Formula
In mathematics, the Riemann–Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a ''ramified covering'' of the other. It therefore connects ramification with algebraic topology, in this case. It is a prototype result for many others, and is often applied in the theory of Riemann surfaces (which is its origin) and algebraic curves. Statement For a compact, connected, orientable surface S, the Euler characteristic \chi(S) is :\chi(S)=2-2g, where ''g'' is the genus (the ''number of handles''). This follows, as the Betti numbers are 1, 2g, 1, 0, 0, \dots. For the case of an (''unramified'') covering map of surfaces :\pi\colon S' \to S that is surjective and of degree N, we have the formula :\chi(S') = N\cdot\chi(S). That is because each simplex of S should be covered by exactly N in S', at least if we use a fine enough triangulation of S, as we are entitled to do since the Euler ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Genus Of A Curve
In mathematics, genus (: genera) has a few different, but closely related, meanings. Intuitively, the genus is the number of "holes" of a surface. A sphere has genus 0, while a torus has genus 1. Topology Orientable surfaces The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It is equal to the number of handles on it. Alternatively, it can be defined in terms of the Euler characteristic \chi, via the relationship \chi=2-2g for closed surfaces, where g is the genus. For surfaces with b boundary components, the equation reads \chi=2-2g-b. In layman's terms, the genus is the number of "holes" an object has ("holes" interpreted in the sense of doughnut holes; a hollow sphere would be considered as having zero holes in this sense). A torus has 1 such hole, while a sphere has 0. The green surface pictured above has 2 holes of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Line
In projective geometry and mathematics more generally, a projective line is, roughly speaking, the extension of a usual line by a point called a '' point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; for example, two distinct projective lines in a projective plane meet in exactly one point (there is no "parallel" case). There are many equivalent ways to formally define a projective line; one of the most common is to define a projective line over a field ''K'', commonly denoted P1(''K''), as the set of one-dimensional subspaces of a two-dimensional ''K''-vector space. This definition is a special instance of the general definition of a projective space. The projective line over the reals is a manifold; see '' Real projective line'' for details. Homogeneous coordinates An arbitrary point in the projective line P1(''K'') may be represented by an equivalence class of '' homogeneous coordi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 dominant if one (equivalently, every) representative f_U in the equivalence class is a dominant morphism, i.e. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacob Lüroth
Jacob Lüroth (18 February 1844, Mannheim, German Confederation, Germany – 14 September 1910, Munich, German Empire, Germany) was a German mathematician who proved Lüroth's theorem and introduced Lüroth quartics. His name is sometimes written Lueroth, following the common Umlaut (diacritic)#Printing conventions in German, printing convention for umlauted characters. He began his studies in astronomy at the University of Bonn, but switched to mathematics when his poor eyesight made taking astronomical observations impossible. He received his doctorate in 1865 from Heidelberg University, for a thesis on Pascal's theorem. From 1868 he was at the Karlsruhe Institute of Technology, where he became a professor in 1869, and from 1880 he was a professor at the Technical University of Munich, succeeding Felix Klein. In 1883, he became a professor at the University of Freiburg, where he remained until his retirement. Following up on Carl Friedrich Gauss's work on statistics, Lüroth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field Extension
In mathematics, particularly in algebra, a field extension is a pair of fields K \subseteq L, such that the operations of ''K'' are those of ''L'' restricted to ''K''. In this case, ''L'' is an extension field of ''K'' and ''K'' is a subfield of ''L''. 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 the multiplicative identity 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Action (mathematics)
In mathematics, a group action of a group G on a set (mathematics), set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformation (function), transformations form a group (mathematics), group under function composition; for example, the rotation (mathematics), rotations around a point in the plane. It is often useful to consider the group as an abstract group, and to say that one has a group action of the abstract group that consists of performing the transformations of the group of transformations. The reason for distinguishing the group from the transformations is that, generally, a group of transformations of a mathematical structure, structure acts also on various related structures; for example, the above rotation group also acts on triangles by transforming triangles into triangles. If a group acts on a structure, it will usually also act on objects built from that st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fixed Point (mathematics)
In mathematics, a fixed point (sometimes shortened to fixpoint), also known as an invariant point, is a value that does not change under a given transformation (mathematics), transformation. Specifically, for function (mathematics), functions, a fixed point is an element that is mapped to itself by the function. Any set of fixed points of a transformation is also an invariant set. Fixed point of a function Formally, is a fixed point of a function if belongs to both the domain of a function, domain and the codomain of , and . In particular, cannot have any fixed point if its domain is disjoint from its codomain. If is defined on the real numbers, it corresponds, in graphical terms, to a curve in the Euclidean plane, and each fixed-point corresponds to an intersection of the curve with the line , cf. picture. For example, if is defined on the real numbers by f(x) = x^2 - 3 x + 4, then 2 is a fixed point of , because . Not all functions have fixed points: for example, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
