Fermat quintic threefold explained

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+Z⁵⁼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:-\zetax:ay:by:cy)

for

\zeta⁵⁼¹

and

a^5+b^5+c⁵⁼⁰

. There are 375 lines in more than one family, of the form

(x:-\zetax:y:-ηy:0)

for fifth roots of unity

\zeta

and