|
Segre Cubic
In algebraic geometry, the Segre cubic is a cubic threefold embedded in 4 (or sometimes 5) dimensional projective space, studied by . Definition The Segre cubic is the set of points (''x''0:''x''1:''x''2:''x''3:''x''4:''x''5) of ''P''5 satisfying the equations :\displaystyle x_0+x_1+x_2+x_3+x_4+x_5= 0 :\displaystyle x_0^3+x_1^3+x_2^3+x_3^3+x_4^3+x_5^3 = 0. Properties The intersection of the Segre cubic with any hyperplane ''x''''i'' = 0 is the Clebsch cubic surface. Its intersection with any hyperplane ''x''''i'' = ''x''''j'' is Cayley's nodal cubic surface. Its dual is the Igusa quartic 3-fold in P4. Its Hessian is the Barth–Nieto quintic. A cubic hypersurface in ''P''4 has at most 10 nodes, and up to isomorphism the Segre cubic is the unique one with 10 nodes. Its nodes are the points conjugate to (1:1:1:−1:−1:−1) under permutations of coordinates. The Segre cubic is rational and furthermore birationally equivalent to a compactification of the Siegel mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cubic Threefold
In algebraic geometry, a cubic threefold is a hypersurface of degree 3 in 4-dimensional projective space. Cubic threefolds are all unirational, but used intermediate Jacobians to show that non-singular cubic threefolds are not rational. The space of lines on a non-singular cubic 3-fold is a Fano surface. Examples *Koras–Russell cubic threefold *Klein cubic threefold *Segre cubic In algebraic geometry, the Segre cubic is a cubic threefold embedded in 4 (or sometimes 5) dimensional projective space, studied by . Definition The Segre cubic is the set of points (''x''0:''x''1:''x''2:''x''3:''x''4:''x''5) of ''P''5 satisfyin ... References * * *{{Citation , last1=Murre , first1=J. P. , author-link1=Jaap Murre , title=Algebraic equivalence modulo rational equivalence on a cubic threefold , url=http://www.numdam.org/item?id=CM_1972__25_2_161_0 , mr=0352088 , year=1972 , journal=Compositio Mathematica , issn=0010-437X , volume=25 , pages=161–206 Algebraic varieties ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally, an affine space with points at infinity, in such a way that there is one point at infinity of each direction of parallel lines. This definition of a projective space has the disadvantage of not being isotropic, having two different sorts of points, which must be considered separately in proofs. Therefore, other definitions are generally preferred. There are two classes of definitions. In synthetic geometry, ''point'' and ''line'' are primitive entities that are related by the incidence relation "a point is on a line" or "a line passes through a point", which is subject to the axioms of projective geometry. For some such set of axioms, the projective spaces that are defined have been shown to be equivalent to those resulting from th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clebsch Cubic Surface
In mathematics, the Clebsch diagonal cubic surface, or Klein's icosahedral cubic surface, is a non-singular cubic surface, studied by and , all of whose 27 exceptional lines can be defined over the real numbers. The term Klein's icosahedral surface can refer to either this surface or its blowup at the 10 Eckardt points. Definition The Clebsch surface is the set of points (''x''0:''x''1:''x''2:''x''3:''x''4) of P4 satisfying the equations :x_0 + x_1 + x_2 + x_3 + x_4 = 0, :x_0^3 + x_1^3 + x_2^3 + x_3^3 + x_4^3 = 0. Eliminating ''x''0 shows that it is also isomorphic to the surface :x_1^3 + x_2^3 + x_3^3 + x_4^3 = (x_1 + x_2 + x_3 + x_4)^3 in P3. Properties The symmetry group of the Clebsch surface is the symmetric group ''S''5 of order 120, acting by permutations of the coordinates (in ''P''4). Up to isomorphism, the Clebsch surface is the only cubic surface with this automorphism group. The 27 exceptional lines are: * The 15 images (under ''S''5) of the line of points o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cayley's Nodal Cubic Surface
In algebraic geometry, the Cayley surface, named after Arthur Cayley, is a cubic nodal surface in 3-dimensional projective space with four conical points. It can be given by the equation : wxy+ xyz+ yzw+zwx =0\ when the four singular points are those with three vanishing coordinates. Changing variables gives several other simple equations defining the Cayley surface. As a del Pezzo surface of degree 3, the Cayley surface is given by the linear system of cubics in the projective plane passing through the 6 vertices of the complete quadrilateral. This contracts the 4 sides of the complete quadrilateral to the 4 nodes of the Cayley surface, while blowing up its 6 vertices to the lines through two of them. The surface is a section through the Segre cubic. The surface contains nine lines, 11 tritangents and no double-sixes. A number of affine forms of the surface have been presented. Hunt uses (1-3 x-3y-3z)(xy+xz+yz)+6xyz = 0 by transforming coordinates (u_0, u_1, u_2, u_3) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Igusa Quartic
In algebraic geometry, the Igusa quartic (also called the Castelnuovo–Richmond quartic ''CR''4 or the Castelnuovo–Richmond–Igusa quartic) is a quartic hypersurface in 4-dimensional projective space, studied by . It is closely related to the moduli space of genus 2 curves with level 2 structure. It is the dual of the Segre cubic In algebraic geometry, the Segre cubic is a cubic threefold embedded in 4 (or sometimes 5) dimensional projective space, studied by . Definition The Segre cubic is the set of points (''x''0:''x''1:''x''2:''x''3:''x''4:''x''5) of ''P''5 satisfyin .... It can be given as a codimension 2 variety in ''P''5 by the equations :\sum x_i=0 :\big(\sum x_i^2\big)^2 = 4 \sum x_i^4 References * * * 3-folds {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barth–Nieto Quintic
In algebraic geometry, the Barth–Nieto quintic is a quintic 3-fold in 4 (or sometimes 5) dimensional projective space studied by that is the Hessian of the Segre cubic. Definition The Barth–Nieto quintic is the closure of the set of points (''x''0:''x''1:''x''2:''x''3:''x''4:''x''5) of P5 satisfying the equations :\displaystyle x_0+x_1+x_2+x_3+x_4+x_5= 0 :\displaystyle x_0^+x_1^+x_2^+x_3^+x_4^+x_5^ = 0. Properties The Barth–Nieto quintic is not rational, but has a smooth model that is a modular Calabi–Yau manifold with Kodaira dimension zero. Furthermore, it is birationally equivalent to a compactification of the Siegel modular variety In mathematics, a Siegel modular variety or Siegel moduli space is an algebraic variety that parametrizes certain types of abelian varieties of a fixed dimension. More precisely, Siegel modular varieties are the moduli spaces of principally pola ... ''A1,3(2)''. References * 3-folds {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 :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 order 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 parameterizat ... [...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 fro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Siegel Modular Variety
In mathematics, a Siegel modular variety or Siegel moduli space is an algebraic variety that parametrizes certain types of abelian varieties of a fixed dimension. More precisely, Siegel modular varieties are the moduli spaces of principally polarized abelian varieties of a fixed dimension. They are named after Carl Ludwig Siegel, the 20th-century German number theorist who introduced the varieties in 1943. Siegel modular varieties are the most basic examples of Shimura varieties. Siegel modular varieties generalize moduli spaces of elliptic curves to higher dimensions and play a central role in the theory of Siegel modular forms, which generalize classical modular forms to higher dimensions. They also have applications to black hole entropy and conformal field theory. Construction The Siegel modular variety ''A''''g'', which parametrize principally polarized abelian varieties of dimension ''g'', can be constructed as the complex analytic spaces constructed as the quotient of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business international ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
