Resolutions of ideals of 6 fat points in the plane.

Let Z be a fat point subscheme of P2 supported at any 6 or fewer distinct points. Although there are infinitely many configurations of the points up to equivalence under the general linear group, if we equate two sets of points, {p1, ..., p6} and {p'1, ..., p'6}, whenever the hilbert function of the ideal I(Z) is the same as that of I(Z') for every choice of nonnegative integers m1, ..., m6, where Z is the fat point subscheme Z = m1p1 + ... + m6p6 and Z' = m1p'1 + ... + m6p'6, then there are only eleven configurations, shown schematically in the figure below. (It turns out that the graded Betti numbers of I(Z) and I(Z') must also be the same.) In particular, configuration 1 consists of 6 general points, for configuration 2 three are collinear, etc. For configuration 11, all 6 are on an irreducible conic. (When it matters which point is assigned which multiplicity, the points are labeled so you know when you use the web form below which point gets which multiplicity mi).

Figure showing all configurations goes here.

To obtain the hilbert function and graded Betti numbers of the ideal I(m1p1 + ... + m6p6), first select which configuration you want to do the computation for, then enter the multiplicities m1, ..., m6. 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 function and graded Betti numbers for any set of six actual fat points in P2 (over an algebraically closed field of arbitrary characteristic) where the fat points have the given configuration type and multiplicities. (In particular, the script does not do a Grobner basis calculation of any kind, and so it is in principle incredibly fast. It does do some string processing in order to apply the theorem, and for speed of coding rather than speed of execution the script was written in awk. Because awk is slow, it can time out if you choose multiplicities which are too big. You can however visit http://www.math.unl.edu/~bharbourne1/6ptres/ and click on "Res6pointDistPts" to download the awk script to run on your own machine. You could also recode it in C and obtain an improvement in execution time, likely better by several orders of magnitude.)

Configuration: Multiplicities: m1= m2= m3= m4= m5= m6=

Explanation of the output

The output for configuration 10 with multiplicities mi = 3 for all i is shown below in red, followed by an explanation of what the output means.

Here is the configuration you entered:
10
Here are the multiplicities you entered:
3, 3, 3, 3, 3, 3
..........................................
Configuration: 10
..........................................
 
The exceptional curves are:
    0  1  0  0  0  0  0
    0  0  1  0  0  0  0
    0  0  0  1  0  0  0
    0  0  0  0  1  0  0
    0  0  0  0  0  1  0
    0  0  0  0  0  0  1
    1 -1  0  0  0  0 -1
    1  0 -1  0  0 -1  0
    1  0  0 -1 -1  0  0
The other negative curves are:
    1 -1 -1 -1  0  0  0
    1 -1  0  0 -1 -1  0
    1  0 -1  0 -1  0 -1
    1  0  0 -1  0 -1 -1
Scheme: Z = 3 3 3 3 3 3
alpha: 7
FCfree in degree: 9
tau: 8
deg    hilb_I  gens    syz    hilb_Z
7       4       4      0       32
8       9       0      3       36
9       19       4      0       36
10       30       0      4       36
11       42       0      0       36
12       55       0      0       36
..........................................