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Glossary Of Classical Algebraic Geometry
The terminology of algebraic geometry changed drastically during the twentieth century, with the introduction of the general methods, initiated by David Hilbert and the Italian school of algebraic geometry in the beginning of the century, and later formalized by André Weil, Jean-Pierre Serre and Alexander Grothendieck. Much of the classical terminology, mainly based on case study, was simply abandoned, with the result that books and papers written before this time can be hard to read. This article lists some of this classical terminology, and describes some of the changes in conventions. translates many of the classical terms in algebraic geometry into scheme-theoretic terminology. Other books defining some of the classical terminology include , , , , , . Conventions The change in terminology from around 1948 to 1960 is not the only difficulty in understanding classical algebraic geometry. There was also a lot of background knowledge and assumptions, much of which has now ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad range of fundamental ideas including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, and the foundations of mathematics (particularly proof theory). He adopted and defended Georg Cantor's set theory and transfinite numbers. In 1900, he presented a collection of problems that set a course for mathematical research of the 20th century. Hilbert and his students contributed to establishing rigor and developed important tools used in modern mathematical physics. He was a cofounder of proof theory and mathematical logic. Life Early life and education Hilbert, the first of two children and only son of O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cross-ratio
In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points , , , on a line, their cross ratio is defined as : (A,B;C,D) = \frac where an orientation of the line determines the sign of each distance and the distance is measured as projected into Euclidean space. (If one of the four points is the line's point at infinity, then the two distances involving that point are dropped from the formula.) The point is the harmonic conjugate of with respect to and precisely if the cross-ratio of the quadruple is , called the ''harmonic ratio''. The cross-ratio can therefore be regarded as measuring the quadruple's deviation from this ratio; hence the name ''anharmonic ratio''. The cross-ratio is preserved by linear fractional transformations. It is essentially the only projective invariant of a quadruple of collinear points; this underlies ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canonizant
In mathematical invariant theory, the canonizant or canonisant is a covariant of forms related to a canonical form for them. Canonizants of a binary form The canonizant of a binary form of degree 2''n'' – 1 is a covariant of degree ''n'' and order ''n'', given by the catalecticant of the penultimate emanant, which is the determinant of the ''n'' by ''n'' Hankel matrix In linear algebra, a Hankel matrix (or catalecticant matrix), named after Hermann Hankel, is a rectangular matrix in which each ascending skew-diagonal from left to right is constant. For example, \qquad\begin a & b & c & d & e \\ b & c & d & e & ... with entries ''a''''i''+''j''''x'' + ''a''''i''+''j''+1''y'' for 0 ≤ ''i'',''j'' < ''n''. References * Invariant theory {{math-stub ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canonical Bundle
In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the nth exterior power of the cotangent bundle \Omega on V. Over the complex numbers, it is the determinant bundle of the holomorphic cotangent bundle T^*V. Equivalently, it is the line bundle of holomorphic n-forms on V. This is the dualising object for Serre duality on V. It may equally well be considered as an invertible sheaf. The canonical class is the divisor class of a Cartier divisor K on V giving rise to the canonical bundle — it is an equivalence class for linear equivalence on V, and any divisor in it may be called a canonical divisor. An anticanonical divisor is any divisor −K with K canonical. The anticanonical bundle is the corresponding inverse bundle \omega^. When the anticanonical bundle of V is ample, V is called a Fano variety. The adjunction formula Suppose that X is a smooth variety and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bitangent
In geometry, a bitangent to a curve is a line that touches in two distinct points and and that has the same direction as at these points. That is, is a tangent line at and at . Bitangents of algebraic curves In general, an algebraic curve will have infinitely many secant lines, but only finitely many bitangents. Bézout's theorem implies that an algebraic plane curve with a bitangent must have degree at least 4. The case of the 28 bitangents of a quartic was a celebrated piece of geometry of the nineteenth century, a relationship being shown to the 27 lines on the cubic surface. Bitangents of polygons The four bitangents of two disjoint convex polygons may be found efficiently by an algorithm based on binary search in which one maintains a binary search pointer into the lists of edges of each polygon and moves one of the pointers left or right at each steps depending on where the tangent lines to the edges at the two pointers cross each other. This bitangent calculati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biregular Map
In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a regular function. A regular map whose inverse is also regular is called biregular, and the biregular maps are the isomorphisms of algebraic varieties. Because regular and biregular are very restrictive conditions – there are no non-constant regular functions on projective varieties – the concepts of rational and birational maps are widely used as well; they are partial functions that are defined locally by rational fractions instead of polynomials. An algebraic variety has naturally the structure of a locally ringed space; a morphism between algebraic varieties is precisely a morphism of the underlying locally ringed spaces. Definition If ''X'' and ''Y'' are closed subvarieties of \mathbb^n and \mathbb^m (so they are affine varieties) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birational Map
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Form
Binary form is a musical form in 2 related sections, both of which are usually repeated. Binary is also a structure used to choreograph dance. In music this is usually performed as A-A-B-B. Binary form was popular during the Baroque music, Baroque period, often used to structure movements of keyboard instrument, keyboard sonata (music), sonatas. It was also used for short, one-movement works. Around the middle of the 18th century, the form largely fell from use as the principal design of entire movements as sonata form and organic musical development, development gained prominence. When it is found in later works, it usually takes the form of the theme in a set of variation form, variations, or the Minuet, Scherzo, or Trio sections of a "minuet and trio" or "scherzo and trio" movement in a sonata, symphony, etc. Many larger forms incorporate binary structures, and many more complicated forms (such as the 18th-century sonata form) share certain characteristics with binary form. S ... [...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] |
Hyperelliptic Surface
In mathematics, a hyperelliptic surface, or bi-elliptic surface, is a minimal surface whose Albanese morphism is an elliptic fibration without singular fibres. Any such surface can be written as the quotient of a product of two elliptic curves by a finite abelian group. Hyperelliptic surfaces form one of the classes of surfaces of Kodaira dimension 0 in the Enriques–Kodaira classification. Invariants The Kodaira dimension is 0. Hodge diamond: Classification Any hyperelliptic surface is a quotient (''E''×''F'')/''G'', where ''E'' = C/Λ and ''F'' are elliptic curves, and ''G'' is a subgroup of ''F'' (acting on ''F'' by translations), which acts on ''E'' not only by translations. There are seven families of hyperelliptic surfaces as in the following table. Here ω is a primitive cube root of 1 and i is a primitive 4th root of 1. Quasi hyperelliptic surfaces A quasi-hyperelliptic surface is a surface whose canonical divisor is numerically equivalent to zero, the Albanes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bicorn
In geometry, the bicorn, also known as a cocked hat curve due to its resemblance to a bicorne, is a Rational curve, rational quartic plane curve, quartic curve defined by the equation y^2 \left(a^2 - x^2\right) = \left(x^2 + 2ay - a^2\right)^2. It has two cusp (singularity), cusps and is symmetric about the y-axis. History In 1864, James Joseph Sylvester studied the curve y^4 - xy^3 - 8xy^2 + 36x^2y+ 16x^2 -27x^3 = 0 in connection with the classification of quintic equations; he named the curve a bicorn because it has two cusps. This curve was further studied by Arthur Cayley in 1867. Properties The bicorn is a algebraic curve, plane algebraic curve of degree four and geometric genus, genus zero. It has two cusp singularities in the real plane, and a double point in the complex projective plane at (x=0, z=0). If we move x=0 and z=0 to the origin and perform an Imaginary number, imaginary rotation on x by substituting ix/z for x and 1/z for y in the bicorn curve, we obtain \left( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bicircular Curve
In geometry, a circular algebraic curve is a type of plane algebraic curve determined by an equation ''F''(''x'', ''y'') = 0, where ''F'' is a polynomial with real coefficients and the highest-order terms of ''F'' form a polynomial divisible by ''x''2 + ''y''2. More precisely, if ''F'' = ''F''''n'' + ''F''''n''−1 + ... + ''F''1 + ''F''0, where each ''F''''i'' is homogeneous of degree ''i'', then the curve ''F''(''x'', ''y'') = 0 is circular if and only if ''F''''n'' is divisible by ''x''2 + ''y''2. Equivalently, if the curve is determined in homogeneous coordinates by ''G''(''x'', ''y'', ''z'') = 0, where ''G'' is a homogeneous polynomial, then the curve is circular if and only if ''G''(1, ''i'', 0) = ''G''(1, −''i'', 0) = 0. In other words, the curve is circular if it contains the circular points at infinity, (1, ''i'', 0) and (1, −''i'',&nb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
