Lines on hypersurfaces -- an example

There are d+1 conditions for a line to be contained in a general hypersurface of degree d in $\PP^n$. The Grassmannian of lines in $\PP^n$ 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 $\PP^4$, 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 $\PP^3$.

We first construct an abstract variety representing the Grassmannian of lines in $\PP^3$ 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 $\PP^3$, that is, an element of the third symmetric power of the space of linear forms, which is the trivial rank 4 bundle on $\PP^3$. Its image in the third symmetric power $Symm^3 Q$ 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

i5 : integral c

o5 = 27

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 0.00354194 seconds
     -- used 0.00478207 seconds
     -- used 0.00738408 seconds
     -- used 0.0220705 seconds
     -- used 0.0355111 seconds
     -- used 0.0623319 seconds
     -- used 0.115571 seconds
     -- used 0.192898 seconds
     -- used 0.305492 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 $\PP^3$, 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.

Lines on hypersurfaces -- an example

See also