Fermat quintic threefold

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

$V^5+W^5+X^5+Y^5+Z^5=0$.

This threefold, so named after Pierre de Fermat, is a Calabi–Yau manifold.

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

Rational curves

Template:Harvs 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 Template:Harvs showed that its lines are contained in 50 1-dimensional families of the form

$(x:-\zeta x:ay:by:cy)$

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

$(x:-\zeta x:y:-\eta y:0)$

for fifth roots of unity $\zeta$ and $\eta$.

References

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