# 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 essentially distinct points (thus infinitely near points are allowed), but where we require the surface X obtained by blowing up the points be such that -KX is nef. Although there are infinitely many configurations of the points up to equivalence under the general linear group, if we equate two sets of points, {A, ..., F} and {A', ..., F'}, whenever the graded Betti numbers of the ideal I(Z) are the same as those of I(Z') for every choice of nonnegative integers m1, ..., m6, where Z is the fat point subscheme Z = m1A + ... + m6F and Z' = m1A' + ... + m6F', then there are 90 configurations, given explicitly in the table below.

Here is what the table means. Configuration 37 is listed as: 2A1A2c 0: AB, DE; 1: ABC, DEF. The expression "2A1A2c" is a unique identifier for the configuration; its meaning is discussed below. The first part, 2A1A2, of the identifier is not unique; several configurations can have the same first part. Thus a small letter, in this case "c", is added to make it unique. This "c" has no meaning except to distinguish between the various configurations which would otherwise all be indicated by "2A1A2". The letters A, ..., F in the expression "0: AB, DE; 1: ABC, DEF" refer to the 6 points. The expression "0: AB, DE" means that point B is infinitely near point A, and point E is infinitely near point D. The expression "1: ABC, DEF" means that, in addition, points A, B and C are collinear, and points D, E and F are collinear. When one sees "2: ABCDEF", this means that points A, B, C, D, E and F lie on an irreducible conic. (From another perspective, this data tells you what the prime divisors of self-intersection -2 are on X. The expression "2: ABCDEF" means that 2L-E1-E2-E3-E4-E5-E6 is the class of a prime divisor, while "1: ABC, DEF" means that L-E1-E2-E3 and L-E4-E5-E6 are, and "0: AB, DE" means that E1-E2 and E4-E5 are. The 0 in "0: AB, DE", the 1 in "1: ABC, DEF" and the 2 in "2: ABCDEF" refer to the coefficient of L in the corresponding divisor class, where: L is the pullback to X of the class of a line in P2; E1 is the class of the scheme-theoretic fiber over the point A under the morphism X->P2 given by blowing up the points A, ..., F; E2 is the class of the fiber over the point B, etc. The expression "2A1A2" in "2A1A2c" is Dynkin diagram notation for the intersection graph of the prime divisors of self-intersection -2. Alternately, the linear system |-KX| defines a morphism f:X->Y of X to a normal cubic surface Y in P3. Then "2A1A2" means that Y has 2 singularities of type A1 and a type A2 singularity. The prime divisors of self intersection -2 are just the components of the curves obtained by minimally resolving the singularities.)

To explain the output, here is the output for the case of configuration 32 with mi=3 for each i:

Here is the configuration you entered:
37
Here are the multiplicities you entered:
3, 3, 3, 3, 3, 3
..........................................
Configuration: 37
..........................................

The exceptional curves are:
0  0  1  0  0  0  0
0  0  0  1  0  0  0
0  0  0  0  0  1  0
0  0  0  0  0  0  1
1 -1  0  0 -1  0  0
1 -1  0  0  0  0 -1
1  0  0 -1 -1  0  0
1  0  0 -1  0  0 -1
The other negative curves are:
0  1 -1  0  0  0  0
0  0  0  0  1 -1  0
1 -1 -1 -1  0  0  0
1  0  0  0 -1 -1 -1
Scheme: Z = 3 3 3 3 3 3
alpha: 6
FCfree in degree: 9
tau: 9
deg    hilb_I  gens    syz    hilb_Z
6       1       1      0       27
7       4       1      0       32
8       10       1      0       35
9       19       1      1       36
10       30       0      1       36
11       42       0      1       36
12       55       0      0       36
..........................................

Your input is echoed, followed by a list of the (-1)-curves on the surface X obtained by blowing up the 6 points, followed by the classes of prime divisors of self-intersection less than -1. The multiplicities are again listed, followed by alpha (the least degree such that the ideal I(Z) in that degree is not 0), then by FCfree (the least degree such that the gcd of the ideal I(Z) in that degree is 1), then by tau (tau+1 is the regularity of I(Z)). Then comes a table showing the values of the Hilbert function hilb_I of the ideal I(Z) and Hilbert function hilb_Z of the quotient ring R/I(Z), where R is the homogeneous coordinate ring of P2. Values are listed only for degrees from alpha to tau+2, since hilb_Z(t) = deg(Z) for t >= tau, and hilb_I(t) = 0 for t < alpha. Note that hilb_I(t) + hilb_Z(t) = t+2 choose 2 for all t. The values given for gen and syz are the graded Betti numbers for the module of generators and module of syzygies.

To obtain the resolution of the ideal I(m1A + ... + m6F), 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/~bharbourne1/6ptsNef-K/ and click on "Res6pointNEF-K" 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.)

Enter the number of the desired configuration:

Enter the desired multiplicities of the points:
m1=
m2=
m3=
m4=
m5=
m6=