Hilbert functions of symbolic powers of n < 9 points in the plane.
Let I(Z) be the ideal defining the subscheme Z
of P2, where Z consists of any n < 9 reduced points.
This web form prints out the hilbert function of k[x,y,z]/I(Z)
for every configuration of n points of multiplicity m.
(Only m < 21 is allowed for this web form. You can
get results for larger m by downloading the program file
that produces the output and running it on your own computer.
An awk version of the file is
at: http://www.math.unl.edu/~bharbourne1/nptsmultm/nptsmultm .
A C version is at:
The awk version is better tested, but it's slower so I'm
using the C version for the website.)
The script does not do a calculation involving an ideal
for some actual set of points; it applies a theorem.
Thus the result does not give experimental data; it gives the actual hilbert functions for the various configurations.
(In particular, the script does not do
a Grobner basis calculation of any kind, and so it is in
fast. The awk version does some string processing in order to apply the theorem,
and for speed of coding rather than speed of execution the script was first written in awk. The C version is much faster; for n=8, m=10 it's
about 200 times faster.)
Note that not every configuration can occur over every
algebraically closed field. Sometimes it depends on the characteristic.
For n=8, there are three configurations that do not occur over any field.
These are configurations 30, 45 and 96. But I show what the
hilbert function would be if they did occur.
For n <= 2, there is only one configuration.
For n = 3, there are two, listed in the following order. The first consists of 3 general points,
the second of three collinear points.
For n = 4, there are three, listed in the following order: four general points;
three points on a line and one off;
and all four on a line.
For n = 5, there are five, listed in the following order: five general points;
three points on a line and two more general points;
four on a line and one off;
all five on a line; and
two points on each of two lines with the fifth point
where the lines meet.
For n=6, 7 or 8, visit the following sites to see the configurations.
The output given here lists the configurations in the same order
as on the pages linked to below:
The first line of output echos your input, say:
"Number of points 8 Multiplicity 1".
The lines after that will look like:
Configs: 132, 135, 136, 137; 1 3 6 7 8 8 8 8 8
This means configurations 132, 135, 136, 137 (and no others)
have the same hilbert function, and this hilbert function
(i.e., the dimension of k[x,y,z]/I(Z) in degree i for each i >= 0)
is 1 in degree 0, 3 in degree 1, 6 in degree 2, etc.,
and the output continues for each degree in turn
until the maximum value of the hilbert function
is reached a few times; that maximum value is always nm(m+1)/2.