In mathematics, a Fermat quintic threefold is a special quintic threefold, in other words a degree 5 dimension 3 hypersurface in 4-dimensional projective space, given by
It is a Calabi–Yau manifold.
The Hodge diamond of a non-singular quintic 3-fold is
Rational curves
Clemens (1984) 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 Albano & Katz (1991) showed that its lines are contained in 50 1-dimensional families of the form (x : −ζx : ay : by : cy) for ζ 5 = 1 and a5 + b5 + c5 = 0. There 375 lines in more than 1 family, of the form (x : −ζx : y : −ηy : 0) for 5th roots of 1 ζ and η.
References
Fermat quintic threefold Wikipedia(Text) CC BY-SA