.......................................... Configuration: 17 .......................................... The exceptional curves are: 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 -1 0 0 -1 0 0 1 0 0 -1 0 0 -1 0 1 0 0 0 -1 0 0 -1 1 -1 0 0 0 -1 0 0 1 -1 0 0 0 0 -1 0 1 -1 0 0 0 0 0 -1 The other negative curves are: 1 -1 -1 -1 -1 0 0 0 1 0 0 0 -1 -1 -1 0 1 0 0 -1 0 -1 0 -1 1 0 -1 0 0 0 -1 -1 Scheme: Z = 3 3 3 3 3 3 3 alpha: 7 FCfree in degree: 12 tau: 11 deg hilb_I lbgens ubgens lbsyz ubsyz hilb_Z 7 2 2 2 0 0 34 8 6 0 0 0 0 39 9 15 4 4 1 1 40 10 25 0 0 4 4 41 11 36 0 0 0 0 42 12 49 1 1 0 0 42 13 63 0 0 1 1 42 14 78 0 0 0 0 42 15 94 0 0 0 0 42 ..........................................Your input is echoed, followed by a list of the (-1)-curves on the surface obtained X by blowing up the 7 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 lbsyz are obtained by taking into account a contribution from the fixed part of the ideal in each degree, and assuming maximal rank for the free part of the ideal in each given degree. The upper bounds ubsyz 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: abcdefg |
3 | 1: abcdef |
4 | 1: abcde |
5 | 1: abcd |
6 | 1: abc |
7 | 1: abcde, afg |
8 | 1: abcd, efg |
9 | 1: abcd, defg |
10 | 1: abcd, def |
11 | 1: abc, def |
12 | 1: abc, ade |
13 | 1: abcd, def, ceg |
14 | 1: abc, def, adg |
15 | 1: abc, ade, afg |
16 | 1: abc, adf, cef |
17 | 1: abcd, def, ceg, bfg |
18 | 1: abc, def, adg, beg |
19 | 1: abc, adf, cef, aeg |
20 | 1: abc, adf, cef, bde |
21 | 1: abc, def, adg, beg, cfg |
22 | 1: abc, adf, cef, bde, aeg |
23 | 1: abc, adf, cef, bde, aeg, cdg |
24 | 1: abc, adf, cef, bde, aeg, cdg, bfg |
25 | 2: abcdefg |
26 | 2: abcdef |
27 | 1: abg; 2: abcdef |
28 | 1: abg, cdg; 2: abcdef |
29 | 1: abg, cdg, efg; 2: abcdef |