Computer code for HMSV paper
This page contains computer code that carries out the calculations needed in the paper ``The equations for the moduli space of n points on the line,'' by Benjamin Howard, John Millson, Andrew Snowden and Ravi Vakil. All the calculations concern the projective coordinate ring for the moduli space of eight points on the projective line.
Proof of Proposition 2.9
The proof of Proposition 2.9 in the paper is a computation done in the
computer algebra system Magma using the sciprt
To run this script simply type
magma simples_generate.mat the command line. The output should look something like this:
Magma V2.13-14 Sun Sep 23 2007 20:19:06 on math [Seed = 3914903004] Type ? for help. Type
-D to quit. Loading file "simples_generate.m" performing degree 2 computation... performing degree 3 computation... performing degree 4 computation (this may take a few minutes)... let I be the ideal of all relations and J the ideal generated by the simple binomial relations and the generalized Segre cubic relation s in degree 2, I has rank 14 and J has rank 14 the largest elementary divisor of the quotient I/J in degree 2 is 1 in degree 3, I has rank 196 and J has rank 196 the largest elementary divisor of the quotient I/J in degree 3 is 1 in degree 4, I has rank 1295 and J has rank 1295 the largest elementary divisor of the quotient I/J in degree 4 is 1 Total time: 23.629 seconds, Total memory usage: 324.82MB
A basis for the quadric relations
We have computed a particularly nice basis for the space of quadric relations on eight points. This basis was not needed in the paper so was not included. It is given by the following relations:
r1 : 0 = hl - ij r2 : 0 = mc - na r3 : 0 = dk - ej r4 : 0 = bf - ga r5 : 0 = ad + aj + di - bc - bh - cl r6 : 0 = jf + ja + fn - hk - hm - kc r7 : 0 = al + aj + le - mb - md - bk r8 : 0 = jc + ja + cg - dh - df - hb r9 : 0 = ak + aj + ki - fm - fl - mh r10 : 0 = jb + ja + bn - ld - lc - dm r11 : 0 = ah + aj + he - cf - ck - fd r12 : 0 = jm + ja + mg - kl - kb - lf r13 : 0 = aj + bh + bj - cg + df + dh + dj + dk + fj - gn + hj + hl + j^2 + jk + jl + jm r14 : 0 = ja + hm + ha - ki + fl + fm + fa + fb + la - ie + ma + mc + a^2 + ab + ac + adThe letters a..n are the generators for the ring given in the paper. This basis has two nice properties. First is that each relation contains a monomial with unit coefficient that does not appear in any other relation (for example,
ijand no other relation does). This implies that the Z-module they generate is saturated and therefore that these relations give an integral basis for the space of all quadratic relations. Second is that they form orbits under the cyclic permutation
(1 2 3 4 5 6 7 8). The first four consitute an orbit as do the next eight and the last two.
We arrived at this basis through a fairly long process. First we used the
rel8 (see below) written by one of us to obtain an
initial basis of the space of quadric relations over Q. We then maniupalted
these relations, essentially by hand, eventually arriving at the above
relations. Finally, we used the magma script
simples_aux.m to verify that these
relations generate the same Z-module as do the simple binomial relations.
Note that the program
rel8 also verifies that the quadric
relations it gives generate the ideal of all relations in degrees 2, 3 and
4 over the rational numbers. Thus we have established this computation
in two independent ways, so we can be fairly certain that an error has not
The source code for
The program computes a basis for I_2 and then calculates the dimension
of the subspaces of I_3 and I_4 generated by I_2 using Matlab. These
dimension calculations show that I_2 does indeed generate I_3 and I_4
over the rationals.
To run this program, first download the source code and deflate the tar
file. Then run the command
make (in the same directory as
the makefile). This should produce an executable file called
rel8. Then run this program.
rel8 invokes Matlab through a system command.
Thus for this program to work you must have Matlab installed and you
must be able to run it from the command line using the command
rel8 will take a minute or two to run. When
it is finished, the screen should look like the following:
initializing program...done outputting degree one generators...done computing matrix of Sym^2(R_1) -> R_2 ...done computing quadratic relations in matlab...done reading output of matlab...done verifying quadratic relations...done outputting quadratic relations...done outputting generated cubic relations...done computing rank in matlab...done outputting generated quartic relations...done computing rank in matlab...done summary of results: R_1 has dimension 14; a basis has been written to deg1_gen I_2 has dimension 14; a basis has been written to quad_rel Sym^3(R_1) has dimension 560; R_3 has dimension 364 thus I_3 has dimension 196 the subspace of I_3 generated by I_2 has dimension 196 therefore I_2 generates I_3 Sym^4(R_1) has dimension 2380; R_4 has dimension 1085 thus I_4 has dimension 1295 the subspace of I_4 generated by I_2 has dimension 1295 therefore I_2 generates I_4The files
quad_relshould have been generated. They should match these files:
Decomposition of the second tensor power
In a previous version of the paper we needed to know how the second
tensor power of the first graded piece of the ring on 8 points
decomposed under the action of the symmetric group
The GAP script
will do this computation.