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


 * $$V^5+W^5+X^5+Y^5+Z^5=0$$.

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

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

Rational curves
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 showed that  its lines are contained in 50 1-dimensional families of the form
 * $$(x : -\zeta x : ay : by : cy) $$

for $$\zeta^5=1$$ and $$a^5+b^5+c^5=0$$. There are 375 lines in more than one family, of the form
 * $$(x : -\zeta x : y :-\eta y :0) $$

for fifth roots of unity $$\zeta$$ and $$\eta$$.