## Preprint

### 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 p_{1}, ..., p_{n}, of P2, and arbitrary
multiplicities m_{1}, ..., m_{n}, the question is to determine
the number of homogeneous generators in each degree for the ideal
I(Z) defining the fat point subscheme Z=m_{1}p_{1}+...+m_{n}p_{n}. 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 m_{i}=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., m_{1}=...=m_{n}),
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 p_{1}, ..., p_{n} of P2
and given m it is not hard to implement a program that
outputs a resolution of I(m_{1}p_{1}+...+m_{n}p_{n}) (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).)