The Ideal Generation Problem for Fat Points

J. Pure Appl. Alg. 145(2), 165--182 (2000).
(This paper extends and makes obsolete my preprint "Generators for Symbolic Powers of Ideals Defining General Points of P2".)

Given general points p1, ..., pn, of P2, and arbitrary multiplicities m1, ..., mn, the question is to determine the number of homogeneous generators in each degree for the ideal I(Z) defining the fat point subscheme Z=m1p1+...+mnpn. Given these numbers, one can write down (up to graded isomorphism) the modules in a minimal free resolution of I(Z).

Unfortunately, not even the Hilbert function of I(Z) is generally known, much less numbers of generators or resolutions. However, the Hilbert function is believed to behave a certain way; in many cases this behavior can be verified, and none are known in which the behavior fails. In this paper, we assume when necessary that this "expected" behavior of the Hilbert function holds.

Now, in the case of thin points (i.e., all mi=1), the resolution is known, and is given in terms of a maximal rank property. This property typically fails for fat points, but the point of the present paper is that the failures seem to be somewhat circumscribed. We show in an appropriate asymptotic sense that maximal rank holds, which therefore allows us to determine a minimal free resolution. In the case of uniform fat point subschemes (i.e., m1=...=mn), we determine all failures of the maximal rank property when n<10, and we conjecture that there are no failures for n>9. We also give some evidence for this conjecture.

In fact, given n general points p1, ..., pn of P2 and given m it is not hard to implement a program that outputs a resolution of I(m1p1+...+mnpn) (conjectural if n>9). Click here to link to the C source text file of this program, or here to link to a Mac version (stuffed and bin hexed) of the program.

Click here to link to a plainTeX text file of this paper.

If you would like to sample the output of the program mentioned above, enter the information required in the data entry fields below, and click on the submit button. (Note: the version of the program run by entering data below is a modified version that suppresses some of the output of the full program, partly to keep down system demands, and partly to avoid needing to explain aspects of the output concerning technical information about I^(m).)

Enter the number n of points here:
Enter the uniform multiplicity m at each point here: