In mathematics, a Fermat quintic threefold is a special quintic threefold

Quintic threefold

In mathematics, a quintic threefold is a degree 5 dimension 3 hypersurface in 4-dimensional projective space. Non-singular quintic threefolds are Calabi-Yau manifolds.The Hodge diamond of a non-singular quintic 3-fold is-Rational curves:...

, in other words a degree 5 dimension 3 hypersurface

Hypersurface

In geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is a hypersurface...

 in 4-dimensional projective space, given by


It is a Calabi-Yau manifold

Calabi-Yau manifold

A Calabi-Yau manifold is a special type of manifold that shows up in certain branches of mathematics such as algebraic geometry, as well as in theoretical physics...

.

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

			1
		0		0
	0		1		0
1		101		101		1
	0		1		0
		0		0
			1

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 : −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.