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
- .
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 |
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
for and . There are 375 lines in more than one family, of the form
for fifth roots of unity and .
gollark: If you are to actually make bold claims about theoretical physics instead of just paraphrasing random quantum things it would be beneficial to learn the relevant maths so you can understand the models.
gollark: I'm glad you are adding topic labels to this. This is very useful and I'd never have known this without you mentioning it.
gollark: Fascinating.
gollark: As far as I know this is only true of some things, and in general things just have mass because they have energy and ??? things occur with this.
gollark: If you are saying "the only reason anything has mass is the Highs boson" then according to my very approximate knowledge this is not true.
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, 106, Princeton University Press, pp. 289–304, MR 0756858
- 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
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