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 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 for \zeta^5=1 and. There are 375 lines in more than one family, of the form for fifth roots of unity \zeta and \eta.