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.
Suppose all you know is n and the hilbert function of I(mZ)
for some m. What can you say about the hilbert functions of
I(rZ) for some other r? This problem (for m=1 and r=2) was posed by
Geramita/Migliore/Sabourin in an ArXiv posting (to appear in J Alg).
This web form gives a solution to the problem for any r and m
(for the safety of the departmental server,
m and r are allowed only to be only 20 or less, but the
programs can be downloaded from
and run for any r and m
within the defined range for integers for your version of C).
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.
Note that not every configuration type can occur over every
algebraically closed field. Sometimes it depends on the characteristic.
This web form ignores that issue, so if you have a particular characteristic in mind you will have to ignore hilbert functions
for types that do not occur in the
characteristic you're interested in.
(An ArXiv posting will be made soon by Geramita/Harbourne/Migliore
that indicates which types occur in characteristics.)
To learn more abut configuration types,
is a pdf file of a talk I gave recently.
The easiest way to specify the hilbert function of
I(mZ) is to specify m, n and the type of the points Z.