Quintic threefold


In mathematics, a quintic threefold is a 3-dimensional hypersurface of degree 5 in 4-dimensional projective space. Non-singular quintic threefolds are Calabi–Yau manifolds.
The Hodge diamond of a non-singular quintic 3-fold is
Mathematician Robbert Dijkgraaf said "One number which every algebraic geometer knows is the number 2,875 because obviously, that is the number of lines on a quintic."

Definition

A quintic threefold is a special class of Calabi–Yau manifolds defined by a degree projective variety in. Many examples are constructed as hypersurfaces in, or complete intersections lying in, or as a smooth variety resolving the singularities of another variety.

Hypersurfaces in P4

Recall that a homogeneous polynomial defines a projective variety, or projective scheme,, from the algebra
where is a field, such as. Then, using the Adjunction formula to compute its canonical bundle, we have
hence in order for the variety to be Calabi-Yau, meaning it has a trivial canonical bundle, its degree must be. It is then a Calabi-Yau manifold if in addition this variety is smooth. This can be checked by looking at the zeros of the polynomials
and making sure the set
is empty.

Examples

Fermat Quintic

One of the easiest examples to check of a Calabi-Yau manifold is given by the Fermat quintic threefold, which is defined by the vanishing locus of the polynomial
Computing the partial derivatives of gives the four polynomials
Since the only points where they vanish is given by the coordinate axes in, the vanishing locus is empty since is not a point in.

As a Hodge Conjecture testbed

Another application of the quintic threefold is in the study of the infinitesimal generalized Hodge conjecture where this difficult problem can be solved in this case. In fact, all of the lines on this hypersurface can be found explicitly.

Dwork family of quintic three-folds

Another popular class of examples of quintic three-folds, studied in many contexts, is the Dwork family. One popular study of such a family is from Candelas, De La Ossa, Green, and Parkes, when they discovered mirror symmetry. This is given by the family
pages 123-125
where is a single parameter not equal to a 5-th root of unity. This can be found by computing the partial derivates of and evaluating their zeros. The partial derivates are given by
At a point where the partial derivatives are all zero, this gives the relation. For example, in we get
by dividing out the and multiplying each side by. From multiplying these families of equations together we have the relation
showing a solution is either given by an or. But in the first case, these give a smooth sublocus since the varying term in vanishes, so a singular point must lie in. Given such a, the singular points are then of the form
such that
where. For example, the point
is a solution of both and its partial derivatives since, and.

Other examples

Computing the number of rational curves of degree can be computed explicitly using Schubert calculus. Let be the rank vector bundle on the Grassmannian of -planes in some rank vector space. Projectivizing to gives the projective grassmannian of degree 1 lines in and descends to a vector bundle on this projective Grassmannian. It's total chern class is
in the Chow ring. Now, a section of the bundle corresponds to a linear homogeneous polynomial,, so a section of corresponds to a quintic polynomial, a section of. Then, in order to calculate the number of lines on a generic quintic threefold, it suffices to compute the integral
This can be done by using the splitting principle. Since
and for a dimension vector space,,
so the total chern class of is given by the product
Then, the euler class, or the top class is
expanding this out in terms of the original chern classes gives
using the relations,.

Rational curves

conjectured that the number of rational curves of a given degree on a generic quintic threefold is finite. This was verified for degrees up to 7 by who also calculated the number 609250 of degree 2 rational curves.
conjectured a general formula for the virtual number of rational curves of any degree, which was proved by .
The number of rational curves of various degrees on a generic quintic threefold is given by
Since the generic quintic threefold is a Calabi–Yau threefold and the moduli space of rational curves of a given degree is a discrete, finite set, these have well-defined Donaldson–Thomas invariants ; at least for degree 1 and 2, these agree with the actual number of points.