Английская Википедия:Fermat quintic threefold

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Файл:CalabiYau5.jpg
Two-dimensional cross-section of the Fermat quintic threefold

In mathematics, a Fermat quintic threefold is a special quintic threefold, in other words a degree 5, dimension 3 hypersurface in 4-dimensional complex projective space, given by the equation

<math>V^5+W^5+X^5+Y^5+Z^5=0</math>.

This threefold, so named after Pierre de Fermat, is a Calabi–Yau manifold.

The Hodge diamond of a non-singular quintic 3-fold is Шаблон:Hodge diamond

Rational curves

Шаблон:Harvs conjectured that the number of rational curves of a given degree on a generic quintic threefold is finite. The Fermat quintic threefold is not generic in this sense, and Шаблон:Harvs showed that its lines are contained in 50 1-dimensional families of the form

<math>(x  : -\zeta x : ay : by : cy) </math>

for <math>\zeta^5=1</math> and <math>a^5+b^5+c^5=0</math>. There are 375 lines in more than one family, of the form

<math>(x  : -\zeta x : y :-\eta y :0) </math>

for fifth roots of unity <math>\zeta</math> and <math>\eta</math>.

References

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Источник — http://wikihandbk.com/ruwiki/index.php?title=Английская_Википедия:Fermat_quintic_threefold&oldid=14685655