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-E

To explain the output, here is the output for the case of configuration 32 with m

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

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 |