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Surface Of General Type
In algebraic geometry, a surface of general type is an algebraic surface with Kodaira dimension 2. Because of Algebraic geometry and analytic geometry#Chow.27s theorem, Chow's theorem any compact complex manifold of dimension 2 and with Kodaira dimension 2 will actually be an algebraic surface, and in some sense most surfaces are in this class. Classification Gieseker showed that there is a coarse moduli scheme for surfaces of general type; this means that for any fixed values of the Chern numbers c_1^2, c_2, there is a quasi-projective scheme classifying the surfaces of general type with those Chern numbers. It remains a very difficult problem to describe these schemes explicitly, and there are few pairs of Chern numbers for which this has been done (except when the scheme is empty). There are some indications that these schemes are in general too complicated to write down explicitly: the known upper bounds for the number of components are very large, some components can be red ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects. The fundamental objects of study in algebraic geometry are algebraic variety, algebraic varieties, which are geometric manifestations of solution set, solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are line (geometry), lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscate of Bernoulli, lemniscates and Cassini ovals. These are plane algebraic curves. A point of the plane lies on an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of points of special interest like singular point of a curve, singular p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Burniat Surface
In mathematics, a Burniat surface is one of the surfaces of general type introduced by . Invariants The geometric genus and irregularity are both equal to 0. The Chern number c_1^2 is either 2, 3, 4, 5, or 6. References * * Algebraic surfaces Complex surfaces {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Modular Surface
In mathematics, a Hilbert modular surface or Hilbert–Blumenthal surface is an algebraic surface obtained by taking a quotient of a product of two copies of the upper half-plane by a Hilbert modular group. More generally, a Hilbert modular variety is an algebraic variety obtained by taking a quotient of a product of multiple copies of the upper half-plane by a Hilbert modular group. Hilbert modular surfaces were first described by using some unpublished notes written by David Hilbert about 10 years before. Definitions If ''R'' is the ring of integers of a real quadratic field, then the Hilbert modular group SL2(''R'') acts on the product ''H''×''H'' of two copies of the upper half plane ''H''. There are several birationally equivalent surfaces related to this action, any of which may be called Hilbert modular surfaces: *The surface ''X'' is the quotient of ''H''×''H'' by SL2(''R''); it is not compact and usually has quotient singularities coming from point ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fano Surface
In algebraic geometry, a Fano surface is a surface of general type In algebraic geometry, a surface of general type is an algebraic surface with Kodaira dimension 2. Because of Algebraic geometry and analytic geometry#Chow.27s theorem, Chow's theorem any compact complex manifold of dimension 2 and with Kodaira ... (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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Surface
In algebraic geometry, a branch of 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 ar ..., a rational surface is a surface birationally equivalent to the projective plane, or in other words a rational variety of dimension two. Rational surfaces are the simplest of the 10 or so classes of surface in the Enriques–Kodaira classification of complex surfaces, and were the first surfaces to be investigated. Structure Every non-singular rational surface can be obtained by repeatedly blowing up a minimal rational surface. The minimal rational surfaces are the projective plane and the Hirzebruch surfaces Σ''r'' for ''r'' = 0 or ''r'' ≥ 2. Invariants: The plurigenera are all 0 and the fundamental group is trivial. Hodge diamond: where ''n'' is 0 for the projective plan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K3 Surface
In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with а trivial canonical bundle and irregularity of a surface, irregularity zero. An (algebraic) K3 surface over any field (mathematics), field means a smooth scheme, smooth proper morphism, proper geometrically connected algebraic surface that satisfies the same conditions. In the Enriques–Kodaira classification of surfaces, K3 surfaces form one of the four classes of minimal surfaces of Kodaira dimension zero. A simple example is the Fermat quartic surface x^4+y^4+z^4+w^4=0 in complex projective space, complex projective 3-space. Together with two-dimensional compact complex tori, K3 surfaces are the Calabi–Yau manifolds (and also the hyperkähler manifolds) of dimension two. As such, they are at the center of the classification of algebraic surfaces, between the positively curved del Pezzo surfaces (which are easy to classify) and the negatively curved surfaces of general t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Intersection
In mathematics, an algebraic variety ''V'' in projective space is a complete intersection if the ideal of ''V'' is generated by exactly ''codim V'' elements. That is, if ''V'' has dimension ''m'' and lies in projective space ''P''''n'', there should exist ''n'' − ''m'' homogeneous polynomials: :F_i(X_0,\cdots,X_n), 1\leq i\leq n - m, in the homogeneous coordinates ''X''''j'', which generate all other homogeneous polynomials that vanish on ''V''. Geometrically, each ''F''''i'' defines a hypersurface; the intersection of these hypersurfaces should be ''V''. The intersection of hypersurfaces will always have dimension at least ''m'', assuming that the field of scalars is an algebraically closed field such as the complex numbers. The question is essentially, can we get the dimension down to ''m'', with no extra points in the intersection? This condition is fairly hard to check as soon as the codimension . When then ''V'' is automatically a hypersurface and there is nothin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Castelnuovo Surface
In 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 ar ..., a Castelnuovo surface is a surface of general type such that the canonical bundle is very ample and such that ''c''12 = 3''pg'' − 7. Guido Castelnuovo proved that if the canonical bundle is very ample for a surface of general type then ''c''12 ≥ 3''pg'' − 7. References {{reflist Algebraic surfaces Complex surfaces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torelli Theorem
In mathematics, the Torelli theorem, named after Ruggiero Torelli, is a classical result of algebraic geometry over the complex number field, stating that a non-singular projective algebraic curve ( compact Riemann surface) ''C'' is determined by its Jacobian variety ''J''(''C''), when the latter is given in the form of a principally polarized abelian variety. In other words, the complex torus ''J''(''C''), with certain 'markings', is enough to recover ''C''. The same statement holds over any algebraically closed field. From more precise information on the constructed isomorphism of the curves it follows that if the canonically principally polarized Jacobian varieties of curves of genus \geq 2 are ''k''-isomorphic for ''k'' any perfect field, so are the curves. This result has had many important extensions. It can be recast to read that a certain natural morphism, the period mapping, from the moduli space of curves of a fixed genus, to a moduli space of abelian varietie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Todorov Surface
In algebraic geometry, a Todorov surface is one of a class of surfaces of general type introduced by for which the conclusion of the Torelli theorem In mathematics, the Torelli theorem, named after Ruggiero Torelli, is a classical result of algebraic geometry over the complex number field, stating that a non-singular projective algebraic curve ( compact Riemann surface) ''C'' is determined b ... does not hold. References * * Algebraic surfaces {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barlow Surface
In mathematics, a Barlow surface is one of the complex surfaces discovered by . They are simply connected surfaces of general type with ''pg'' = 0. They are homeomorphic but not diffeomorphic to a projective plane blown up in 8 points. The Hodge diamond for the Barlow surfaces is: See also * Hodge theory References * * * * Algebraic surfaces Complex surfaces {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat Surface
In geometry, the Fermat cubic, named after Pierre de Fermat, is a surface defined by : x^3 + y^3 + z^3 = 1. \ Methods of algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ... provide the following parameterization of Fermat's cubic: : x(s,t) = : y(s,t) = : z(s,t) = . In projective space the Fermat cubic is given by :w^3+x^3+y^3+z^3=0. The 27 lines lying on the Fermat cubic are easy to describe explicitly: they are the 9 lines of the form (''w'' : ''aw'' : ''y'' : ''by'') where ''a'' and ''b'' are fixed numbers with cube −1, and their 18 conjugates under permutations of coordinates. ::::''Real points of Fermat cubic surface.'' References * * Algebraic surfaces {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
