|
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] |
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 of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypersurface
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidean space, an affine space or a projective space. Hypersurfaces share, with surfaces in a three-dimensional space, the property of being defined by a single implicit equation, at least locally (near every point), and sometimes globally. A hypersurface in a (Euclidean, affine, or projective) space of dimension two is a plane curve. In a space of dimension three, it is a surface. For example, the equation :x_1^2+x_2^2+\cdots+x_n^2-1=0 defines an algebraic hypersurface of dimension in the Euclidean space of dimension . This hypersurface is also a smooth manifold, and is called a hypersphere or an -sphere. Smooth hypersurface A hypersurface that is a smooth manifold is called a ''smooth hypersurface''. In , a smooth hypersurface is orienta ... [...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 the fol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unirational
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 parameterizati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intermediate Jacobian
In mathematics, the intermediate Jacobian of a compact Kähler manifold or Hodge structure is a complex torus that is a common generalization of the Jacobian variety of a curve and the Picard variety and the Albanese variety. It is obtained by putting a complex manifold, complex structure on the torus H^n(M,\R)/H^n(M,\Z) for ''n'' odd. There are several different natural ways to put a complex structure on this torus, giving several different sorts of intermediate Jacobians, including one due to and one due to . The ones constructed by Weil have natural polarizations if ''M'' is projective, and so are abelian varieties, while the ones constructed by Griffiths behave well under holomorphic deformations. A complex structure on a real vector space is given by an automorphism ''I'' with square -1. The complex structures on H^n(M,\R) are defined using the Hodge decomposition : H^(M,) \otimes = H^(M)\oplus\cdots\oplus H^(M). On H^ the Weil complex structure I_W is multiplication by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fano Surface
In algebraic geometry, a Fano surface is a surface of general type (in particular, not a Fano variety) whose points index the lines on a non-singular cubic threefold. They were first studied by . Hodge diamond: Fano surfaces are perhaps the simplest and most studied examples of irregular surfaces of general type that are not related to a product of two curves and are not a complete intersection of divisors in an Abelian variety. The Fano surface S of a smooth cubic threefold F into P4 carries many remarkable geometric properties. The surface S is naturally embedded into the grassmannian of lines G(2,5) of P4. Let U be the restriction to S of the universal rank 2 bundle on G. We have the: Tangent bundle Theorem (Fano, Clemens-Griffiths The surname Griffiths is a surname with Welsh origins, as in Gruffydd ap Llywelyn Fawr. People called Griffiths recorded here include: * Alan Griffiths (born 1952), Australian politician and businessman * Alan Griffiths (cricketer) (born 195 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Koras–Russell Cubic Threefold
In algebraic geometry, the Koras–Russell cubic threefolds are smooth affine complex threefolds diffeomorphic to \mathbf^3studied by . They have a hyperbolic action of a one-dimensional torus \mathbf^*with a unique fixed point, such that the quotients of the threefold and the tangent space of the fixed point by this action are isomorphic. They were discovered in the process of proving the Linearization Conjecture in dimension 3. A linear action of \mathbf^* on the affine space \mathbf^n is one of the form t*(x_1,\ldots,x_n)=(t^x_1,t^x_2,\ldots,t^x_n), where a_1,\ldots,a_n\in \mathbf and t\in\mathbf^*. The Linearization Conjecture in dimension n says that every algebraic action of \mathbf^* on the complex affine space \mathbf^n is linear in some algebraic coordinates on \mathbf^n. M. Koras and P. Russell made a key step towards the solution in dimension 3, providing a list of threefolds (now called Koras-Russell threefolds) and proving that the Linearization Conjecture for n=3 ho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Klein Cubic Threefold
In algebraic geometry, the Klein cubic threefold is the non-singular cubic threefold in 4-dimensional projective space given by the equation :V^2W+W^2X+X^2Y+Y^2Z+Z^2V =0 \, studied by . Its automorphism group is the group PSL2(11) of order 660 . It is unirational but not a rational variety. showed that it is birational to the moduli space of (1,11)-polarized abelian surfaces. References * * *{{Citation , authorlink=Felix Klein , last1=Klein , first1=Felix , title=Ueber die Transformation elfter Ordnung der elliptischen Functionen , doi=10.1007/BF02086276 , year=1879 , journal=Mathematische Annalen ''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, ... , issn=0025-5831 , volume=15 , issue=3 , pages=533–555, url=https://zenodo.org/record/1642598 3-folds ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 modular v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. The n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Varieties
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition. Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to be irreducible, which means that it is not the union of two smaller sets that are closed in the Zariski topology. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility. The fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial (an algebraic object) in one variable with complex number coefficients is determined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
