Here is the configuration you entered:

10

Here are the multiplicities you entered:

3, 3, 3, 3, 3, 3

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

- Let X -> P
^{2}be the morphism obtained by blowing up the points p_{i}. The divisor class group of X is a free abelian group with basis L, E_{1}, ..., E_{6}, where L is the pullback of the class of a line in P^{2}, and E_{i}is the class of the inverse image of p_{i}. An exceptional curve is a smooth curve isomorphic to P^{1}whose class C satisfies C^{2}= -1. All of the exceptional curves are listed. The notation "1 -1 0 0 0 0 -1" for an exceptional curve just gives the coefficients of the class of the curve in terms of the basis L, E_{1}, ..., E_{6}, so "1 -1 0 0 0 0 -1" denotes L - E_{1}- E_{6}. All curves of negative self-intersection in fact are isomorphic to P^{1}. The classes of all such curves C with C^{2}< -1 are also listed. - The quantity called alpha is just t+1,
where t is the largest degree such that
I(Z)
_{t}= 0. Thus alpha is the degree in which I(Z) starts being nonzero. - For t large enough, the zero locus (as a point set)
of I(Z)
_{t}is just the set of six points p_{i}, but for smaller values of t the base locus can include other points and even curves. Thus "FCfree in degree: 9" tells us that in degree 9 (but not degree 8) the base locus is 0-dimensional. (It turns out to be true that in the case of 6 points, when the base locus contains no curves, then it contains no points other than the 6 points p_{i}.) - The definition of the hilbert function
hilb
_{I(Z)}in degree t is just that it is the vector space dimension of the homogeneous component I(Z)_{t}of the ideal I(Z) in degree t; i.e., hilb_{I(Z)}(t) = dim I(Z)_{t}. The ideal I(Z) is an ideal of the ring R=k[x,y,z]. The quotient R/I(Z) is thus a graded R-module, so we can take its hilbert function, which we refer to as hilb_{Z}. The two hilbert functions are related by the formula hilb_{I(Z)}(t) + hilb_{Z}(t) = (t+2)(t+1)/2 for t >= 0. Clearly, the hilbert function of the ideal I(Z) is 0 in degrees less than alpha, so there is no point in listing its values for t < alpha. It turns out that the hilbert function hilb_{Z}of Z increases from 1 to a constant. The degree t in which this constant is first achieved is called tau. Clearly there is no point in listing the hilbert function of Z in degrees greater than tau. So the output gives a table of values of both hilbert functions for degrees t between alpha and tau. The column denoted "gens" gives the number of generators in each degree t in a minimal set of homogeneous generators for I(Z). The column listed "syz" does the same for the module of syzygies. These are just the graded Betti numbers for I(Z). The last generator of I(Z) always occurs in degree at most tau + 1 and the last syzygy in degree at most tau + 3. The table continues past tau to a degree or two past the degree of the last generator and the last syzygy.