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Beauville Surface
In mathematics, a Beauville surface is one of the surfaces of general type introduced by . They are examples of "fake quadrics", with the same Betti numbers as quadric surfaces. Construction Let ''C''1 and ''C''2 be smooth curves with genera ''g''1 and ''g''2. Let ''G'' be a finite group acting on ''C''1 and ''C''2 such that *''G'' has order (''g''1 − 1)(''g''2 − 1) *No nontrivial element of ''G'' has a fixed point on both ''C''1 and ''C''2 *''C''1/''G'' and ''C''2/''G'' are both rational. Then the quotient (''C''1 × ''C''2)/''G'' is a Beauville surface. One example is to take ''C''1 and ''C''2 both copies of the genus 6 quintic ''X''5 + ''Y''5 + ''Z''5 =0, and ''G'' to be an elementary abelian group of order 25, with suitable actions on the two curves. Invariants Hodge diamond Homological mirror symmetry is a mathematical conjecture made by Maxim Kontsevich. It seeks a systematic mathematical explanation for a phenomenon called mirror symmetry first observed by p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surfaces 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] |
Betti Number
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicial complexes or CW complexes), the sequence of Betti numbers is 0 from some point onward (Betti numbers vanish above the dimension of a space), and they are all finite. The ''n''th Betti number represents the rank of the ''n''th homology group, denoted ''H''''n'', which tells us the maximum number of cuts that can be made before separating a surface into two pieces or 0-cycles, 1-cycles, etc. For example, if H_n(X) \cong 0 then b_n(X) = 0, if H_n(X) \cong \mathbb then b_n(X) = 1, if H_n(X) \cong \mathbb \oplus \mathbb then b_n(X) = 2, if H_n(X) \cong \mathbb \oplus \mathbb\oplus \mathbb then b_n(X) = 3, etc. Note that only the ranks of infinite groups are considered, so for example if H_n(X) \cong \mathbb^k \oplus \mathbb/(2) , where \mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hodge Diamond
Homological mirror symmetry is a mathematical conjecture made by Maxim Kontsevich. It seeks a systematic mathematical explanation for a phenomenon called mirror symmetry first observed by physicists studying string theory. History In an address to the 1994 International Congress of Mathematicians in Zürich, speculated that mirror symmetry for a pair of Calabi–Yau manifolds ''X'' and ''Y'' could be explained as an equivalence of a triangulated category constructed from the algebraic geometry of ''X'' (the derived category of coherent sheaves on ''X'') and another triangulated category constructed from the symplectic geometry of ''Y'' (the derived Fukaya category). Edward Witten originally described the topological twisting of the N=(2,2) supersymmetric field theory into what he called the A and B model topological string theories. These models concern maps from Riemann surfaces into a fixed target—usually a Calabi–Yau manifold. Most of the mathematical predictions o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Surfaces
In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of dimension four as a smooth manifold. The theory of algebraic surfaces is much more complicated than that of algebraic curves (including the compact Riemann surfaces, which are genuine surfaces of (real) dimension two). Many results were obtained, however, in the Italian school of algebraic geometry, and are up to 100 years old. Classification by the Kodaira dimension In the case of dimension one varieties are classified by only the topological genus, but dimension two, the difference between the arithmetic genus p_a and the geometric genus p_g turns to be important because we cannot distinguish birationally only the topological genus. Then we introduce the irregularity for the classification of them. A summary of the results (in det ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
