Trans. Amer. Math. Soc., 2002

Resolutions of Ideals of Quasiuniform Fat Point Subschemes of P2

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 P2 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, mth symbolic powers of an ideal defining n=r2>9 general points in P2, 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 P2, 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.]
Enter the number n of points here:
Enter the uniform multiplicity m at each point here: