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 http://www.math.unl.edu/~bharbourne1/GMS/ 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, here 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.




Number of points:

Multiplicity m:

The configuration type must be at least 1 and up to:
1 for n=1 or n=2 points,
2 for n=3,
3 for n=4,
5 for n=5,
11 for n=6,
29 for n=7, and
146 for n=8.
Enter the desired configuration type of the points (note: types 30, 45 and 96 are never representable so have been excluded from the list):

Multiplicity r: