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 m_i).

Figure showing all configurations goes here.

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 m_i = 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
..........................................

Let X -> P² be the morphism obtained by blowing up the points p_i. The divisor class group of X is a free abelian group with basis L, E₁, ..., E₆, where L is the pullback of the class of a line in P², and E_i is the class of the inverse image of p_i. An exceptional curve is a smooth curve isomorphic to P¹ whose class C satisfies C² = -1. All of the exceptional curves are listed. The notation "1 -1 0 0 0 0 -1" for an exceptional curve just gives the coefficients of the class of the curve in terms of the basis L, E₁, ..., E₆, so "1 -1 0 0 0 0 -1" denotes L - E₁ - E₆. All curves of negative self-intersection in fact are isomorphic to P¹. The classes of all such curves C with C² < -1 are also listed.
The quantity called alpha is just t+1, where t is the largest degree such that I(Z)_t = 0. Thus alpha is the degree in which I(Z) starts being nonzero.
For t large enough, the zero locus (as a point set) of I(Z)_t is just the set of six points p_i, but for smaller values of t the base locus can include other points and even curves. Thus "FCfree in degree: 9" tells us that in degree 9 (but not degree 8) the base locus is 0-dimensional. (It turns out to be true that in the case of 6 points, when the base locus contains no curves, then it contains no points other than the 6 points p_i.)
The definition of the hilbert function hilb_I(Z) in degree t is just that it is the vector space dimension of the homogeneous component I(Z)_t of the ideal I(Z) in degree t; i.e., hilb_I(Z)(t) = dim I(Z)_t. The ideal I(Z) is an ideal of the ring R=k[x,y,z]. The quotient R/I(Z) is thus a graded R-module, so we can take its hilbert function, which we refer to as hilb_Z. The two hilbert functions are related by the formula hilb_I(Z)(t) + hilb_Z(t) = (t+2)(t+1)/2 for t >= 0. Clearly, the hilbert function of the ideal I(Z) is 0 in degrees less than alpha, so there is no point in listing its values for t < alpha. It turns out that the hilbert function hilb_Z of Z increases from 1 to a constant. The degree t in which this constant is first achieved is called tau. Clearly there is no point in listing the hilbert function of Z in degrees greater than tau. So the output gives a table of values of both hilbert functions for degrees t between alpha and tau. The column denoted "gens" gives the number of generators in each degree t in a minimal set of homogeneous generators for I(Z). The column listed "syz" does the same for the module of syzygies. These are just the graded Betti numbers for I(Z). The last generator of I(Z) always occurs in degree at most tau + 1 and the last syzygy in degree at most tau + 3. The table continues past tau to a degree or two past the degree of the last generator and the last syzygy.