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 formula_1 projective variety in formula_2. Many examples are constructed as hypersurfaces in formula_2, or complete intersections lying in formula_2, or as a smooth variety resolving the singularities of another variety. As a set, a Calabi-Yau manifold isformula_5where formula_6 is a degree formula_1 homogeneous polynomial. One of the most studied examples is from the polynomialformula_8called 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.
Recall that a homogeneous polynomial formula_9 (where formula_10 is the Serre-twist of the hyperplane line bundle) defines a projective variety, or projective scheme, formula_11, from the algebraformula_12where formula_13 is a field, such as formula_14. Then, using the Adjunction formula to compute its canonical bundle, we haveformula_15hence in order for the variety to be Calabi-Yau, meaning it has a trivial canonical bundle, its degree must be formula_1. 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 polynomialsformula_17and making sure the setformula_18is 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 polynomialformula_19Computing the partial derivatives of formula_20 gives the four polynomialsformula_21Since the only points where they vanish is given by the coordinate axes in formula_2, the vanishing locus is empty since formula_23 is not a point in formula_2.
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 123-125formula_25where formula_26 is a single parameter not equal to a 5-th root of unity. This can be found by computing the partial derivates of formula_27 and evaluating their zeros. The partial derivates are given byformula_28At a point where the partial derivatives are all zero, this gives the relation formula_29. For example, in formula_30 we getformula_31by dividing out the formula_1 and multiplying each side by formula_33. From multiplying these families of equations formula_29 together we have the relationformula_35showing a solution is either given by an formula_36 or formula_37. But in the first case, these give a smooth sublocus since the varying term in formula_27 vanishes, so a singular point must lie in formula_37. Given such a formula_26, the singular points are then of the formformula_41 such that formula_42where formula_43. For example, the pointformula_44is a solution of both formula_45 and its partial derivatives since formula_46, and formula_47.
Curves on a quintic threefold.
Computing the number of rational curves of degree formula_48 can be computed explicitly using Schubert calculus. Let formula_49 be the rank formula_50 vector bundle on the Grassmannian formula_51 of formula_50-planes in some rank formula_1 vector space. Projectivizing formula_51 to formula_55 gives the projective grassmannian of degree 1 lines in formula_2 and formula_49 descends to a vector bundle on this projective Grassmannian. Its total chern class isformula_58in the Chow ring formula_59. Now, a section formula_60 of the bundle corresponds to a linear homogeneous polynomial, formula_61, so a section of formula_62 corresponds to a quintic polynomial, a section of formula_63. Then, in order to calculate the number of lines on a generic quintic threefold, it suffices to compute the integralformula_64This can be done by using the splitting principle. Sinceformula_65and for a dimension formula_50 vector space, formula_67,formula_68so the total chern class of formula_62 is given by the productformula_70Then, the Euler class, or the top class isformula_71expanding this out in terms of the original chern classes givesformula_72using the relations formula_73, formula_74.
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. 
conjectured a general formula for the virtual number of rational curves of any degree, which was proved by (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 
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.