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Quartic Threefold
In algebraic geometry, a quartic threefold is a degree 4 hypersurface of dimension 3 in 4-dimensional projective space. showed that all non-singular quartic threefolds are irrational, though some of them are unirational. Examples *Burkhardt quartic * Igusa quartic References *{{Citation , last1=Iskovskih , first1=V. A. , last2=Manin , first2=Ju. I. , title=Three-dimensional quartics and counterexamples to the Lüroth problem , doi= 10.1070/SM1971v015n01ABEH001536 , mr=0291172 , year=1971 , journal=Matematicheskii Sbornik ''Matematicheskii Sbornik'' (russian: Математический сборник, abbreviated ''Mat. Sb.'') is a peer reviewed Russian mathematical journal founded by the Moscow Mathematical Society in 1866. It is the oldest successful Russian mat ... , series=Novaya Seriya , volume=86 , pages=140–166 3-folds ... [...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] |
Burkhardt Quartic
In mathematics, the Burkhardt quartic is a quartic threefold in 4-dimensional projective space studied by , with the maximum possible number of 45 nodes. Definition The equations defining the Burkhardt quartic become simpler if it is embedded in ''P''5 rather than ''P''4. In this case it can be defined by the equations σ1 = σ4 = 0, where σ''i'' is the ''i''th elementary symmetric function of the coordinates (''x''0 : ''x''1 : ''x''2 : ''x''3 : ''x''4 : ''x''5) of ''P''5. Properties The automorphism group of the Burkhardt quartic is the Burkhardt group ''U''4(2) = PSp4(3), a simple group of order 25920, which is isomorphic to a subgroup of index 2 in the Weyl group of E6. The Burkhardt quartic is rational and furthermore birationally equivalent to a compactification of the Siegel modular variety In mathematics, a Siegel modular variety or Siegel moduli space is an algebraic variety that parametrizes certain types of abelian varieties of a fixed dimension. More precisely, Sie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Igusa Quartic
In algebraic geometry, the Igusa quartic (also called the Castelnuovo–Richmond quartic ''CR''4 or the Castelnuovo–Richmond–Igusa quartic) is a quartic hypersurface in 4-dimensional projective space, studied by . It is closely related to the moduli space of genus 2 curves with level 2 structure. It is the dual of the 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 .... It can be given as a codimension 2 variety in ''P''5 by the equations :\sum x_i=0 :\big(\sum x_i^2\big)^2 = 4 \sum x_i^4 References * * * 3-folds {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matematicheskii Sbornik
''Matematicheskii Sbornik'' (russian: Математический сборник, abbreviated ''Mat. Sb.'') is a peer reviewed Russian mathematical journal founded by the Moscow Mathematical Society in 1866. It is the oldest successful Russian mathematical journal. The English translation is ''Sbornik: Mathematics''. It is also sometimes cited under the alternative name ''Izdavaemyi Moskovskim Matematicheskim Obshchestvom'' or its French translation ''Recueil mathématique de la Société mathématique de Moscou'', but the name ''Recueil mathématique'' is also used for an unrelated journal, '' Mathesis''. Yet another name, ''Sovetskii Matematiceskii Sbornik'', was listed in a statement in the journal in 1931 apologizing for the former editorship of Dmitri Egorov, who had been recently discredited for his religious views; however, this name was never actually used by the journal. The first editor of the journal was Nikolai Brashman, who died before its first issue (dedicated to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |