Two-dimensional cross-section of the 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

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

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

Herbert 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 Alberto Albano and Sheldon Katz (1991) showed that its lines are contained in 50 1-dimensional families of the form

${\displaystyle (x:-\zeta x:ay:by:cy)}$

for ${\displaystyle \zeta ^{5}=1}$ and ${\displaystyle a^{5}+b^{5}+c^{5}=0}$. There are 375 lines in more than one family, of the form

${\displaystyle (x:-\zeta x:y:-\eta y:0)}$

for fifth roots of unity ${\displaystyle \zeta }$ and ${\displaystyle \eta }$.

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), Annals of Mathematics Studies, vol. 106, Princeton University Press, pp. 289–304, MR 0756858
• Cox, David A.; Katz, Sheldon (1999), Mirror symmetry and algebraic geometry, Mathematical Surveys and Monographs, vol. 68, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1059-0, MR 1677117