Hilbert functions of ideals of n < 9 fat points in the plane.

Let I(Z) be the ideal defining the subscheme Z of P2, where Z is a fat point scheme supported at any given configuration of n < 9 points where the points are assigned multiplicities m_1, ..., m_n. This web form prints out the hilbert function of k[x,y,z]/I(Z). (Only m_i < 21 is allowed for this web form. You can get results for larger m by downloading and compiling the program file that produces the output and running it on your own computer. A C version is at: http://www.math.unl.edu/~bharbourne1/hilbfuncts/hilbfuncts.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 for the various configurations. (In particular, the script does not do a Grobner basis calculation of any kind, and so it is in principle incredibly fast. Using it over the web can involve network delays.)

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 the input allows you to choose these configurations. The algorithm which outputs the hilbert function can still be run for these cases; the output shows it gives.

For n <= 2, there is only one configuration.

For n = 3, there are two, listed in the following order. Choose configuration 1 to get 3 general points, or choose configuration 2 for three collinear points.

For n = 4, there are three, with 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, ordered as follows: 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:

n=6, n=7, n=8




Choose the verbosity of the output:

Choose the number n of points:

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:

Now choose the multiplicities:
Multiplicity m1: Multiplicity m2: Multiplicity m3: Multiplicity m4:

Multiplicity m5: Multiplicity m6: Multiplicity m7: Multiplicity m8:

Explanation of the output

The output first echos your input. If you selected a verbosity of 0 then the lines after that will look like:
Config 4: 1 3 6 9 12 15 18 20 22 24 26 27 28 29 30 30 30 30 30
This means for configuration 4, the value of the hilbert function of k[x,y,z]/I(Z) in degree 0 is 1, in degree 1 it's 3, etc., and the output continues for each degree in turn until the maximum value of the hilbert function is reached; that maximum value is always \sum_i m_i(m_i+1)/2. If your verbosity is 1 then the output also includes a list of all curves of negative self-intersection for the specified configuration type, the degree in which the ideal of the fat points is first fixed component free, and then the values of the hilbert function of the ideal I(Z) and the quotient k[x,y,z]/I(Z) are listed in each degree starting from the last degree for which the hilbert function of I(Z) is 0 up to a few past the first degree in which the hilbert function of k[x,y,z]/I(Z) attains its maximum value.