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 projective space, given by

${\displaystyle V^{5}+W^{5}+X^{5}+Y^{5}+Z^{5}=0.\,}$

It is a Calabi–Yau manifold.

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

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

• Albano, Alberto; Katz, Sheldon (1991), "Lines on the Fermat quintic threefold and the infinitesimal generalized Hodge conjecture", Transactions of the American Mathematical Society, 324 (1): 353–368, doi:10.2307/2001512, ISSN 0002-9947, JSTOR 2001512, MR 1024767
• Clemens, Herbert (1984), "Some results about Abel-Jacobi mappings", Topics in transcendental algebraic geometry (Princeton, N.J., 1981/1982), Ann. of Math. Stud., 106, Princeton University Press, pp. 289–304, MR 756858
• Cox, David A.; Katz, Sheldon (1999), Mirror symmetry and algebraic geometry, Mathematical Surveys and Monographs, 68, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1059-0, MR 1677117