There are d+1 conditions for a line to be contained in a general hypersurface of degree d in ℙn. The Grassmannian of lines in ℙn has dimension 2(n-1). Therefore, when d+1 = 2(n-1), we should expect a finite number of lines. Here is a way of computing the number using Schubert2. In the case of lines on a quintic hypersurface in ℙ4, this computation was done by Hermann Schubert in 1879.
We will first illustrate the method by computing the number of lines on a cubic surface in ℙ3.
We first construct
an abstract variety representing the Grassmannian of lines in
ℙ3 and then retrieve its tautological sub- and quotient bundles.
i1 : G = flagBundle({2,2}, VariableNames => {,c})
o1 = G
o1 : a flag bundle with subquotient ranks {2:2}
|
i2 : (S,Q) = bundles G
o2 = (S, Q)
o2 : Sequence
|
Any cubic surface is given by a cubic form on ℙ3, that is, an element of the third symmetric power of the space of linear forms, which is the trivial rank 4 bundle on ℙ3. Its image in the third symmetric power Symm3 Q of the quotient bundle Q vanishes at those points of the Grassmannian that correspond to lines on which the cubic form vanishes identically, that is, lines contained in the cubic surface. The class of this locus is the top Chern class of this bundle.
i3 : B = symmetricPower(3,Q)
o3 = B
o3 : an abstract sheaf of rank 4 on G
|
i4 : c = chern(rank B,B)
2
o4 = 27c
2
QQ[][H , H , c , c ]
1,1 1,2 1 2
o4 : ---------------------------------------------------------------
(- H - c , - H - H c - c , - H c - H c , -H c )
1,1 1 1,2 1,1 1 2 1,2 1 1,1 2 1,2 2
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i5 : integral c
o5 = 27
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We can do the same thing for any n, (with d = 2n-3) as follows:
i6 : f = n -> (
G := flagBundle({n-1,2});
integral chern symmetricPower_(2*n-3) last bundles G
)
o6 = f
o6 : FunctionClosure
|
i7 : for n from 2 to 10 list time f n
-- used 0.0182037 seconds
-- used 0.0262804 seconds
-- used 0.0449957 seconds
-- used 0.0803438 seconds
-- used 0.32519 seconds
-- used 0.236162 seconds
-- used 0.501017 seconds
-- used 0.759103 seconds
-- used 1.38188 seconds
o7 = {1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775,
------------------------------------------------------------------------
289139638632755625, 520764738758073845321}
o7 : List
|
Note: in characteristic zero, using Bertini’s theorem, the numbers computed can be proved to be equal to the actual numbers of distinct lines for general hypersurfaces. In ℙ3, every smooth cubic surface in characteristic zero has exactly 27 lines. In higher dimensions there may be smooth hypersurfaces for which the number of lines is different from the “expected” number that we have computed above. For example, the Fermat quintic threefold has an infinite number of lines on it.