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From Wikipedia, the free encyclopedia
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
1 | ||||||
0 | 0 | |||||
0 | 1 | 0 | ||||
1 | 101 | 101 | 1 | |||
0 | 1 | 0 | ||||
0 | 0 | |||||
1 |
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
for and . There are 375 lines in more than one family, of the form
for fifth roots of unity and .
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