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Catanese Surface
In mathematics, a Catanese surface is one of the surfaces of general type introduced by . Construction The construction starts with a quintic ''V'' with 20 double points. Let ''W'' be the surface obtained by blowing up the 20 double points. Suppose that ''W'' has a double cover ''X'' branched over the 20 exceptional −2-curves. Let ''Y'' be obtained from ''X'' by blowing down the 20 −1-curves in ''X''. If there is a group of order 5 acting freely on all these surfaces, then the quotient ''Z'' of ''Y'' by this group of order 5 is a Catanese surface. Catanese found a 4-dimensional family of curves constructed like this. Invariants The Catanese surface is a numerical Campedelli surface and hence has Hodge diamond and canonical degree K^2 = 2. The fundamental group of the Catanese surface is \mathbf/5\mathbf, as can be seen from its quotient construction. References * *{{Citation , last1=Catanese , first=Fabrizio, authorlink=Fabrizio Catanese , title=Babbage's conje ... [...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] |
Campedelli Surface
In mathematics, a Campedelli surface is one of the surfaces of general type introduced by Campedelli. Surfaces with the same Hodge number In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every cohom ...s are called numerical Campedelli surfaces. Construction Invariants Hodge diamond: References *{{Citation , last1=Barth , first1=Wolf P. , last2=Hulek , first2=Klaus , last3=Peters , first3=Chris A.M. , last4=Van de Ven , first4=Antonius , title=Compact Complex Surfaces , publisher= Springer-Verlag, Berlin , series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. , isbn=978-3-540-00832-3 , mr=2030225 , year=2004 , volume=4 Algebraic surfaces Complex surfaces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homological Mirror Symmetry
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 of mi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inventiones Mathematicae
''Inventiones Mathematicae'' is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current managing editors are Camillo De Lellis (Institute for Advanced Study, Princeton) and Jean-Benoît Bost (University of Paris-Sud Paris-Sud University (French: ''Université Paris-Sud''), also known as University of Paris — XI (or as Université d'Orsay before 1971), was a French research university distributed among several campuses in the southern suburbs of Paris, in ...). Abstracting and indexing The journal is abstracted and indexed in: References External links *{{Official website, https://www.springer.com/journal/222 Mathematics journals Publications established in 1966 English-language journals Springer Science+Business Media academic journals Monthly journals ... [...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] |