## Trans. Amer. Math. Soc., 2002

### Resolutions of Ideals of Quasiuniform Fat Point Subschemes of P^{2}

Joint with S. Holay and S. Fitchett

This paper gives the first determination of
the minimal free resolution of the mth symbolic power
of an ideal of n simple general points of P^{2}
for certain n with both m and n arbitrarily. This
is ibtained as a Corollary of a result of L. Evain
and results in this preprint, in which resolutions of fat point ideals
on P2 are determined when the ideals are, for m
sufficiently large, m^{th} symbolic
powers of an ideal defining n=r^{2}>9 general points in P^{2},
under the assumption that the Hilbert function of the symbolic power
takes on a well-known naive minimum value (i.e.,
assuming that the fat points impose independent
conditions in each degree for which the Hilbert
function is nonzero). Results are also
determined for infinitely many m for each n>9 a nonsquare.

## Notes

Because of the assumption that the Hilbert function
takes on the naive minimum value, and because this has not yet
been proved to hold for n>9 general points of P^{2},
resolutions computed under this assumption must be regarded as
conjectural. The following web form can be used to display
these conjectural resolutions. (For n<=9, the form will return
the actual resolution. Note that in this case the Hilbert
function can indeed fail to take on the naive minimum value.)
Cases in which the results returned are conjectural are flagged with a
warning to that effect. Also, our results tying the resolution
to the Hilbert function only hold for values of
m sufficiently large, but we conjecture that the resolution
should behave similarly for all m; thus the form below produces
a result, possibly conjectural, for any m.
[Note: values of m or n bigger than 10000 will not be accepted,
in order not to overburden the server.]