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

Physicist 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. Many examples are constructed as hypersurfaces, or complete intersections homogeneous polynomialcalled a Fermat polynomial. Proving that such a polynomial defines a Calabi-Yau requires some more tools, like the Adjunction formula and conditions for smoothness.

Hypersurfaces in P4

is the Serre-twist of the hyperplane line bundle) defines a projective variety, or projective scheme. Then using the adjunction formula to compute its canonical bundle. It is then a Calabi-Yau manifold if in addition this variety is smoothis empty.

Examples

Fermat Quintic

One of the easiest examples to check of a Calabi-Yau manifold is given by the Fermat quintic threefold.

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 is a single parameter not equal to a 5-th root of unity.

Other examples

• Barth–Nieto quintic

• Consani–Scholten quintic

Curves on a quintic threefold

can be computed explicitly using Schubert calculus vector bundle on the Grassmannian vector space descends to a vector bundle on this projective Grassmannian. Its total Chern classin the Chow ring. Then, in order to calculate the number of lines on a generic quintic threefold, it suffices to compute the integralThis can be done by using the splitting principleThen, the Euler classusing relations implied by Pieri's formula.

Rational curves

conjectured that the number of rational curves of a given degree on a generic quintic threefold is finite. (Some smooth but non-generic quintic threefolds have infinite families of lines on them.) This was verified for degrees up to 7 by who also calculated the number 609250 of degree 2 rational curves. Brian Greene and Ronen Plesser (1989) proposed a construction for a mirror manifold of quintic threefolds. conjectured a general formula for the virtual number of rational curves of any degree. The formula was independently verified by and . The fact that the virtual number equals the actual number relies on confirmation of Clemens' conjecture, currently known for degree at most 11 . The number of rational curves of various degrees on a generic quintic threefold is given by

1. 2875, 609250, 317206375, 242467530000, ....

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 (hence compact), these have well-defined Donaldson–Thomas invariants (the "virtual number of points"); at least for degree 1 and 2, these agree with the actual number of points.

See also

• Deformation theory

• Enumerative geometry

• Gromov–Witten invariant

• Hodge structure

• Jacobian ideal - gives an explicit basis for the Hodge-decomposition

• Mirror symmetry (string theory)

• Schubert calculus - techniques for determining the number of lines on a quintic threefold

References