In mathematics, a Fermat quintic threefold is a special

quintic threefold In mathematics, a quintic threefold is a 3-dimensional hypersurface of degree 5 in 4-dimensional projective space \mathbb^4. Non-singular quintic threefolds are Calabi–Yau manifolds. The Hodge diamond of a non-singular quintic 3-fold is Physi ...

, in other words a degree 5,

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...

hypersurface In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidea ...

in 4-dimensional complex

projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...

, given by the equation :

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

. This threefold, so named after

Pierre de Fermat Pierre de Fermat (; ; 17 August 1601 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his d ...

, is a

Calabi–Yau manifold In algebraic and differential geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has certain properties, such as Ricci flatness, yielding applications in theoretical physics. P ...

. The

Hodge diamond Homological mirror symmetry is a mathematical conjecture made by Maxim Kontsevich. It seeks a systematic mathematical explanation for a phenomenon called mirror symmetry first observed by physicists studying string theory. History In an address ...

of a non-singular quintic 3-fold is

Rational curves

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 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 In mathematics, a root of unity is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group char ...

\zeta

and

\eta

References

* * *{{Citation , last1=Cox , first1=David A. , authorlink1= David A. Cox , last2=Katz , first2=Sheldon , authorlink2= Sheldon Katz , title=Mirror symmetry and algebraic geometry , url=https://www.ams.org/bookstore-getitem/item=surv-68.s , publisher=

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

, location=Providence, R.I. , series=Mathematical Surveys and Monographs , isbn=978-0-8218-1059-0 , mr=1677117 , year=1999 , volume=68 3-folds Complex manifolds