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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] |
Fano Variety
In algebraic geometry, a Fano variety, introduced by Gino Fano in , is a complete variety ''X'' whose anticanonical bundle ''K''X* is ample. In this definition, one could assume that ''X'' is smooth over a field, but the minimal model program has also led to the study of Fano varieties with various types of singularities, such as terminal or klt singularities. Recently techniques in differential geometry have been applied to the study of Fano varieties over the complex numbers, and success has been found in constructing moduli spaces of Fano varieties and proving the existence of Kähler–Einstein metrics on them through the study of K-stability of Fano varieties. Examples * The fundamental example of Fano varieties are the projective spaces: the anticanonical line bundle of P''n'' over a field ''k'' is ''O''(''n''+1), which is very ample (over the complex numbers, its curvature is ''n+1'' times the Fubini–Study symplectic form). * Let ''D'' be a smooth codimension-1 s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surface Of General Type
In algebraic geometry, a surface of general type is an algebraic surface with Kodaira dimension 2. Because of 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 non-reduced everywhere, components may have many different ... [...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] |
Gino Fano
Gino Fano (5 January 18718 November 1952) was an Italian mathematician, best known as the founder of finite geometry. He was born to a wealthy Jewish family in Mantua, in Italy and died in Verona, also in Italy. Fano made various contributions on projective and algebraic geometry. His work in the foundations of geometry predates the similar, but more popular, work of David Hilbert by about a decade. He was the father of physicist Ugo Fano and electrical engineer Robert Fano and uncle to physicist and mathematician Giulio Racah. Mathematical work Fano was an early writer in the area of finite projective spaces. In his article on proving the independence of his set of axioms for projective ''n''-space, among other things, he considered the consequences of having a fourth harmonic point be equal to its conjugate. This leads to a configuration of seven points and seven lines contained in a finite three-dimensional space with 15 points, 35 lines and 15 planes, in which each lin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Herbert Clemens
Charles Herbert Clemens Jr. (born 15 August 1939 in Dayton, Ohio) is an American mathematician, specializing in complex algebraic geometry. Biography Clemens received in 1961 his bachelor's degree from College of the Holy Cross and in 1966 his Ph.D. from the University of California, Berkeley under Phillip Griffiths with thesis ''Picard–Lefschetz Theorem for Families of Algebraic Varieties Acquiring Certain Singularities''. In 1970 he became an assistant professor at Columbia University and went on to become an associate professor before leaving in 1975 to become an associate professor at the University of Utah where he became a full professor in 1976 and a Distinguished Professor in 2001. In 2002 he left Utah to become a professor of mathematics and mathematics education at the Ohio State University. Clemens was a visiting scholar at the Institute for Advanced Study from September 1968 to March 1970 and from September 2001 to June 2003. He was an invited speaker at the Intern ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Phillip Griffiths
Phillip Augustus Griffiths IV (born October 18, 1938) is an American mathematician, known for his work in the field of geometry, and in particular for the complex manifold approach to algebraic geometry. He was a major developer in particular of the theory of variation of Hodge structure in Hodge theory and moduli theory. He also worked on partial differential equations, coauthored with Shiing-Shen Chern, Robert Bryant and Robert Gardner on Exterior Differential Systems. Professional career He received his BS from Wake Forest College in 1959 and his PhD from Princeton University in 1962 after completing a doctoral dissertation, titled "On certain homogeneous complex manifolds", under the supervision of Donald Spencer. Afterwards, he held positions at University of California, Berkeley (1962–1967) and Princeton University (1967–1972). Griffiths was a professor of mathematics at Harvard University from 1972 to 1983. He was then a Provost and James B. Duke Professor o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hodge Theory
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 cohomology class has a canonical representative, a differential form that vanishes under the Laplacian operator of the metric. Such forms are called harmonic. The theory was developed by Hodge in the 1930s to study algebraic geometry, and it built on the work of Georges de Rham on de Rham cohomology. It has major applications in two settings: Riemannian manifolds and Kähler manifolds. Hodge's primary motivation, the study of complex projective varieties, is encompassed by the latter case. Hodge theory has become an important tool in algebraic geometry, particularly through its connection to the study of algebraic cycles. While Hodge theory is intrinsically dependent upon the real and complex numbers, it can be applied to questions ... [...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 ... [...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 detai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
