Hilbert functions of ideals of 8 fat points in the plane.

Let Z be a fat point subscheme of P² supported at any 8 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, {A, ..., H} and {A', ..., H'}, whenever the Hilbert function of the ideal I(Z) is the same as those of I(Z') for every choice of nonnegative integers m₁, ..., m₈, where Z is the fat point subscheme Z = m₁A + ... + m₈H and Z' = m₁A' + ... + m₈H', then there are 146 configurations, given explicitly in the table below.

Here is what the table means. Configuration 37 is listed as: 37 1: ABC, ADE, AFG, 3: ABCDEFGH. The letters A, ..., H in the expression "1: ABC, ADE, AFG, 3: ABCDEFGH" refer to the 8 points. The expression "1: ABC, ADE, AFG" means that points A, B and C are collinear, and points A, D, E are collinear, and A, F and G are collinear. The expression "3: ABCDEFGH" means that points A, B, C, D, E, F, G and H lie on an irreducible cubic which is singular at H. 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 the surface X obtained by blowing up the points. The expression "2: ABCDEF" means that 2L-E₁-E₂-E₃-E₄-E₅-E₆ is the class of a prime divisor, while "1: ABC, ADE" means that L-E₁-E₂-E₃ and L-E₁-E₄-E₅ are, and "3: ABCDEFGH" means that 3L-E₁-...-E₇-2E₈ is. In the last case, the coefficient of -2 is always attached to point H. The 1 in "1: ABC, DEF" and the 2 in "2: ABCDEF" and the 3 in "3: ABCDEFGH" 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, ..., H; E₂ is the class of the fiber over the point B, etc.

To obtain the Hilbert function of the ideal I(m₁A + ... + m₈H), first select which configuration you want to do the computation for, then enter the multiplicities m₁, ..., m_H. 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 for any set of eight 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/~bharbourne1/8ptres/ and click on "Res8point" 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=
m₇=
m₈=

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

Here is the configuration you entered:

..........................................
Configuration: 32
..........................................
 
The exceptional curves are:
    0  1  0  0  0  0  0  0  0
    0  0  1  0  0  0  0  0  0
    0  0  0  1  0  0  0  0  0
    0  0  0  0  1  0  0  0  0
    0  0  0  0  0  1  0  0  0
    0  0  0  0  0  0  1  0  0
    0  0  0  0  0  0  0  1  0
    0  0  0  0  0  0  0  0  1
    1 -1  0  0  0  0  0  0 -1
    1  0 -1  0  0  0  0 -1  0
    1  0  0 -1 -1  0  0  0  0
    1  0  0  0  0 -1 -1  0  0
The other negative curves are:
    1 -1 -1 -1  0  0  0  0  0
    1 -1  0  0 -1 -1  0  0  0
    1 -1  0  0  0  0 -1 -1  0
    1  0 -1  0 -1  0 -1  0  0
    1  0  0 -1  0 -1  0 -1  0
    1  0 -1  0  0 -1  0  0 -1
    1  0  0 -1  0  0 -1  0 -1
    1  0  0  0 -1  0  0 -1 -1
Scheme: Z = 3 3 3 3 3 3 3 3
alpha: 8
FCfree in degree: 9
tau: 9
deg    hilb_I  lbgens  ubgens  lbsyz  ubsyz  hilb_Z
8       1       1      1      0       0       44
9       7       4      4      0       0       48
10       18       0      1      0       1       48
11       30       0      0      4       4       48
12       43       0      0      0       0       48
..........................................

Your input is echoed, followed by a list of the (-1)-curves on the surface obtained X by blowing up the 8 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 table also shows upper and lower bounds on the Betti numbers of the resolution of the ideal (ubgens, lbgens, resp., for numbers of generators, and ubsyz, lbsyz for the syzygies). The lower bounds lbgens are obtained by a maximal rank assumption on the free part of the ideal in a given degree, and the upper bounds ubgens are obtained using bounds involving l and q given in "Resolutions of Ideals of Quasiuniform Fat Point Subschemes of P2", Harbourne, Holay and Fitchett, TAMS, 2002.

1	Empty
2	1: ABC
3	1: ABC, DEF
4	1: ABC, ADE
5	1: ABC, ADE, AFG
6	1: ABC, ADE, BDF
7	1: ABC, ADE, BFG
8	1: ABC, ADE, FGH
9	1: ABC, ADE, BDF, CGH
10	1: ABC, ADE, BDF, CEG
11	1: ABC, ADE, BDF, CEF
12	1: ABC, ADE, BFG, DFH
13	1: ABC, ADE, AFG, BDF
14	1: ABC, ADE, AFG, BDH
15	1: ABC, ADE, AFG, BDF, CEG
16	1: ABC, ADE, AFG, BDF, BEG
17	1: ABC, ADE, AFG, BDF, CEH
18	1: ABC, ADE, AFG, BDF, BEH
19	1: ABC, ADE, AFG, BDH, CEH
20	1: ABC, ADE, AFG, BDH, CFH
21	1: ABC, ADE, BDF, CGH, EFG
22	1: ABC, ADE, AFG, BDF, CEG, BEH
23	1: ABC, ADE, AFG, BDF, BEG, CDG
24	1: ABC, ADE, AFG, BDF, BEG, DGH
25	1: ABC, ADE, AFG, BDF, BEG, CDH
26	1: ABC, ADE, AFG, BDF, CEH, BGH
27	1: ABC, ADE, AFG, BDH, CFH, EGH
28	1: ABC, ADE, AFG, BDF, CEG, BEH, CFH
29	1: ABC, ADE, AFG, BDF, CEG, BEH, CDH
30	1: ABC, ADE, AFG, BDF, BEG, CDG, CEF
31	1: ABC, ADE, AFG, BDF, BEG, CDG, CEH
32	1: ABC, ADE, AFG, BDF, CEG, BEH, CFH, DGH
33	3: ABCDEFGH
34	1: ABC, 3: ABCDEFGH
35	1: ABC, DEF, 3: ABCDEFGH
36	1: ABC, ADE, 3: ABCDEFGH
37	1: ABC, ADE, AFG, 3: ABCDEFGH
38	1: ABC, ADE, BDF, 3: ABCDEFGH
39	1: ABC, ADE, BFG, 3: ABCDEFGH
40	1: ABC, ADE, BDF, CEG, 3: ABCDEFGH
41	1: ABC, ADE, BDF, CEF, 3: ABCDEFGH
42	1: ABC, ADE, AFG, BDF, 3: ABCDEFGH
43	1: ABC, ADE, AFG, BDF, CEG, 3: ABCDEFGH
44	1: ABC, ADE, AFG, BDF, BEG, 3: ABCDEFGH
45	1: ABC, ADE, AFG, BDF, BEG, CDG, 3: ABCDEFGH
46	1: ABC, ADE, AFG, BDF, BEG, CDG, CEF, 3: ABCDEFGH
47	2: ABCDEF, 3: ABCDEFGH
48	1: ABC, 2: BCDEFG, 3: ABCDEFGH
49	1: ABC, ADE, 2: BCDEFG, 3: ABCDEFGH
50	1: ABC, ADE, AFG, 2: BCDEFG, 3: ABCDEFGH
51	2: ABCDEF
52	2: ABCDEF, ABCDGH
53	2: ABCDEF, ABCDGH, ABEFGH
54	2: ABCDEF, ABCDGH, ABEFGH, CDEFGH
55	1: ABC, ADE, FGH, 2: BCDEGH
56	1: ABC, ADE, BDF, CGH, 2: ABEFGH
57	1: ABC, ADE, AFG, BDF, 2: CDEFGH
58	1: ABC, ADE, AFG, BDF, BEG, 2: CDEFGH
59	1: ABC, ADE, AFG, BDF, BEH, 2: CDEFGH
60	1: ABC, ADE, AFG, BDH, CEH, 2: BCDEFG
61	1: ABC, ADE, AFG, BDH, CFH, 2: BCDEFG
62	1: ABC, ADE, AFG, BDH, CFH, EGH, 2: BCDEFG
63	1: ABC, 2: CDEFGH
64	1: ABC, ADE, 2: BCDEGH
65	1: ABC, ADE, AFG, 2: BCDEFG
66	1: ABC, ADE, BDF, 2: CDEFGH
67	1: ABC, ADE, BFG, 2: CDEFGH
68	1: ABC, ADE, AFG, BDH, 2: CDEFGH
69	1: ABC, 2: BCDEFG
70	1: ABC, ADE, 2: CDEFGH
71	1: ABC, ADE, AFG, 2: CDEFGH
72	1: ABC, ADE, BDF, 2: BCDEGH
73	1: ABC, ADE, BFG, 2: BCDEGH
74	1: ABC, ADE, AFG, BDH, 2: BCDEFG
75	1: ABC, ADE, 2: ACEFGH
76	1: ABC, DEF, 2: BCEFGH
77	1: ABC, ADE, BDF, CEG, 2: ACDFGH
78	1: ABC, ADE, BDF, CEF, 2: BCDEGH
79	1: ABC, ADE, BFG, DFH, 2: BCDEGH
80	1: ABC, 2: CDEFGH, ABEFGH
81	1: ABC, ADE, 2: BCDEGH, ACEFGH
82	1: ABC, ADE, BDF, 2: BCDEGH, ACDFGH
83	1: ABC, 2: ACDEFG, ABEFGH
84	1: ABC, ADE, 2: BDEFGH, ACEFGH
85	1: ABC, ADE, BDF, 2: CDEFGH, ABEFGH
86	1: ABC, ADE, 2: ACEFGH, ABDFGH
87	1: ABC, DEF, 2: BCEFGH, ACDFGH
88	1: ABC, ADE, BFG, 2: BCDEGH, ACEFGH
89	1: ABC, ADE, BDF, CEG, 2: ACDFGH, ABEFGH
90	1: ABC, ADE, BDF, CEF, 2: BCDEGH, ACDFGH
91	1: ABC, ADE, BFG, DFH, 2: BCDEGH, ACEFGH
92	1: ABC, 2: BCDFGH, ACDEFG, ABEFGH
93	1: ABC, DEF, 2: BCEFGH, ACDFGH, ABDEGH
94	1: ABC, ADE, 2: BCDEGH, ACEFGH, ABDFGH
95	1: ABC, ADE, BDF, 2: BCDEGH, ACDFGH, ABEFGH
96	1: ABC, ADE, BDF, CEF, 2: BCDEGH, ACDFGH, ABEFGH
97	1: ABC, ADE, AFGH, 2: BCDEGH
98	1: ABC, DEFG
99	1: ABC, ADE, AFGH
100	1: ABC, ADE, BDF, AFGH
101	1: ABC, ADE, AFG, BDF, CEGH
102	1: ABC, ADEH, 2: BCDEFG
103	1: ABC, ADE, BDF, AFGH, 2: BCDEGH
104	1: ABC, ADEF
105	1: ABC, ADE, BDFG
106	1: ABC, ADE, BDF, CEGH
107	1: ABC, ADE, AFG, BDF, BEGH
108	1: ABC, ADE, BDFG, 2: ACEFGH
109	1: ABC, ADE, BFGH
110	1: ABC, ADE, BDF, CEFG
111	1: ABC, DEF, ADGH, 2: BCEFGH
112	1: ABC, ADE, BDF, CEF, AFGH, 2: BCDEGH
113	1: ABCD
114	1: ABC, ADE, AFG, BDFH
115	1: ABCD, EFGH
116	1: ABCD, AEFG
117	1: ABC, ADEF, BDGH
118	1: ABC, ADE, BDFG, CEFH
119	1: ABC, ADE, AFG, BDFH, CEGH
120	1: ABGH, 2: ABCDEF
121	1: EFGH, 2: ABCDEF, ABCDGH
122	1: ABC, DEF, ADGH
123	1: ABC, ADE, BFG, CDFH
124	1: ABC, ADE, BDF, CEG, AFGH
125	1: ABC, ADE, BDF, CEF, AFGH
126	1: ABC, ADE, BFG, DFH, CEGH
127	1: ABC, ADE, AFG, BDH, CEFH
128	1: ABC, ADE, AFG, BDF, BEG, CDGH
129	1: ABC, ADE, AFG, BDF, BEH, CDGH
130	1: ABC, ADE, AFG, BDF, BEG, CDG, CEFH
131	1: ABC, ADE, AFG, BDF, BEG, DGH, CEFH
132	1: ABCDE
133	1: ABCDE, AFGH
134	1: ABCDE, FGH
135	1: ABCDE, AFG
136	1: ABCDE, AFG, BFH
137	1: ABCDE, AFG, BFH, CGH
138	1: ABCDEF
139	1: ABCDEF, AGH
140	1: ABCDEFG
141	1: ABCDEFGH
142	2: ABCDEFG
143	1: ABH, 2: ABCDEFG
144	1: ABH, CDH, 2: ABCDEFG
145	1: ABH, CDH, EFH, 2: ABCDEFG
146	2: ABCDEFGH