Here is what the table means. Configuration 37 is listed as: 2A

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

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

1 | Empty | 46 | A_{1}A_{3}d | 0: AB, BC; 1: ABC, ADE | |

2 | A_{1}a | 0: AB | 47 | A_{1}A_{3}e | 0: AF, BC; 1: ABC, ADE |

3 | A_{1}b | 1: ABC | 48 | A_{1}A_{3}f | 0: BC, CD; 1: ABC, DEF |

4 | A_{1}c | 2: ABCDEF | 49 | A_{1}A_{3}g | 0: BC, CD, EF; 1: ABC |

5 | 2A_{1}a | 0: AB, CD | 50 | A_{1}A_{3}h | 0: AB, BC, CD; 2: ABCDEF |

6 | 2A_{1}b | 0: AB; 1: ABC | 51 | 2A_{2}a | 0: AB, BC, DE, EF |

7 | 2A_{1}c | 0: DE; 1: ABC | 52 | 2A_{2}b | 0: AB, CF; 1: ABC, ADE |

8 | 2A_{1}d | 0: AB; 2: ABCDEF | 53 | 2A_{2}c | 0: AB, BC; 1: ABC, DEF |

9 | 2A_{1}e | 1: ABC, ADE | 54 | A_{4}a | 0: AB, BC, CD, DE |

10 | A_{2}a | 0: AB, BC | 55 | A_{4}b | 0: AB, CD, DE; 1: ACD |

11 | A_{2}b | 0: CD; 1: ABC | 56 | A_{4}c | 0: AB, BC, EF; 1: ADE |

12 | A_{2}c | 1: ABC, DEF | 57 | A_{4}d | 0: CD, DE, EF; 1: ABC |

13 | 3A_{1}a | 0: AB, CD, EF | 58 | A_{4}e | 0: BC, CD; 1: ABC, BEF |

14 | 3A_{1}b | 0: AB, DE; 1: ABC | 59 | A_{4}f | 0: AB, BC, CD; 1: ABC |

15 | 3A_{1}c | 0: BC; 1: ABC, ADE | 60 | D_{4}a | 0: BC, CD, DE; 1: ABC |

16 | 3A_{1}d | 0: AB, CD; 2: ABCDEF | 61 | D_{4}b | 0: AB, CD, EF; 1: ACE |

17 | 3A_{1}e | 1: ABC, ADE, BDF | 62 | A_{1}2A_{2}a | 0: AB, BC, DE, EF; 1: ABC |

18 | A_{1}A_{2}a | 0: AB, BC, DE | 63 | A_{1}2A_{2}b | 0: AB, CF, DE; 1: ABC, ADE |

19 | A_{1}A_{2}b | 0: AB, BC; 1: ABC | 64 | A_{1}2A_{2}c | 0: AB, BC, DE; 1: ABC, DEF |

20 | A_{1}A_{2}c | 0: AB, BC; 1: DEF | 65 | A_{1}2A_{2}d | 0: AB, CD; 1: ABC, AEF, CDE |

21 | A_{1}A_{2}d | 0: AB, CD; 1: ABC | 66 | A_{1}2A_{2}e | 0: AB, BC, DE, EF; 2: ABCDEF |

22 | A_{1}A_{2}e | 0: CD, EF; 1: ABC | 67 | 2A_{1}A_{3}a | 0: BC, CD, EF; 1: ABC, AEF |

23 | A_{1}A_{2}f | 0: CD; 1: ABC, AEF | 68 | 2A_{1}A_{3}b | 0: AD, CE; 1: ABC, ADF, CEF |

24 | A_{1}A_{2}g | 0: AB; 1: ABC, ADE | 69 | 2A_{1}A_{3}c | 0: AB, BC, DE; 1: ABC, ADE |

25 | A_{1}A_{2}h | 0: AB; 1: ABC, DEF | 70 | 2A_{1}A_{3}d | 0: AF, BC, DE; 1: ABC, ADE |

26 | A_{1}A_{2}i | 0: AB, BC; 2: ABCDEF | 71 | 2A_{1}A_{3}e | 0: BC, CF, DE; 1: ABC, ADE |

27 | A_{3}a | 0: AB, BC, CD | 72 | 2A_{1}A_{3}f | 0: AB, BC, CD, EF; 2: ABCDEF |

28 | A_{3}b | 0: CD, DE; 1: ABC | 73 | A_{1}A_{4}a | 0: AB, BC, CD, EF; 1: ABC |

29 | A_{3}c | 0: AB, DE; 1: ACD | 74 | A_{1}A_{4}b | 0: AB, CD, DE, EF; 1: ABC |

30 | A_{3}d | 0: AF; 1: ABC, ADE | 75 | A_{1}A_{4}c | 0: AB, DE, EF; 1: ABC, ADE |

31 | A_{3}e | 0: BC, CD; 1: ABC | 76 | A_{1}A_{4}d | 0: AB, BF, DE; 1: ABC, ADE |

32 | 4A_{1}a | 0: BC, DE; 1: ABC, ADE | 77 | A_{1}A_{4}e | 0: AB, BC, EF; 1: ABC, ADE |

33 | 4A_{1}b | 0: AB, CD, EF; 2: ABCDEF | 78 | A_{1}A_{4}f | 0: AB, BC, CD, DE; 2: ABCDEF |

34 | 4A_{1}c | 1: ABC, ADE, BDF, CEF | 79 | A_{5}a | 0: AB, BC, CD, DE, EF |

35 | 2A_{1}A_{2}a | 0: AB, BC, DE; 1: ABC | 80 | A_{5}b | 0: AB, BC, DE, EF; 1: ADE |

36 | 2A_{1}A_{2}b | 0: AB, CD, EF; 1: ABC | 81 | A_{5}c | 0: AB, BC, CD; 1: ABC, AEF |

37 | 2A_{1}A_{2}c | 0: AB, DE; 1: ABC, DEF | 82 | D_{5}a | 0: BC, CD, DE, EF; 1: ABC |

38 | 2A_{1}A_{2}d | 0: AB, DE, EF; 1: ABC | 83 | D_{5}b | 0: AB, CD, DE, EF; 1: ACD |

39 | 2A_{1}A_{2}e | 0: AB, DE; 1: ABC, ADE | 84 | D_{5}c | 0: AB, BC, CD, DE; 1: ABC |

40 | 2A_{1}A_{2}f | 0: BF, DE; 1: ABC, ADE | 85 | 3A_{2}a | 0: AB, BC, DE, EF; 1: ABC, DEF |

41 | 2A_{1}A_{2}g | 0: AC; 1: ABC, ADE, BDF | 86 | 3A_{2}b | 0: AB, CD, EF; 1: ABC, AEF, CDE |

42 | 2A_{1}A_{2}h | 0: AB, BC, DE; 2: ABCDEF | 87 | A_{1}A_{5}a | 0: AB, BC, DE, EF; 1: ABC, ADE |

43 | A_{1}A_{3}a | 0: AB, BC, CD, EF | 88 | A_{1}A_{5}b | 0: AB, BC, CF, DE; 1: ABC, ADE |

44 | A_{1}A_{3}b | 0: AB, CD, DE; 1: ABC | 89 | A_{1}A_{5}c | 0: AB, BC, CD, DE, EF; 2: ABCDEF |

45 | A_{1}A_{3}c | 0: AB, DF; 1: ABC, ADE | 90 | E_{6} | 0: AB, BC, CD, DE, EF; 1: ABC |