An Algorithm for Fat Points on P2

Given general points p1, ..., pr, of P2, and arbitrary multiplicities m1, ..., mr, 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+...+mrpr. Given these numbers, one can write down (up to graded isomorphism) the modules in a minimal free resolution of I(Z).

It is easy to translate this problem into one concerning line bundles on the blow up of P2 at the points p1, ..., pr. Under reasonable assumptions, this preprint shows that the problem reduces to the case of ample line bundles. As a consequence, we give a complete solution to the problem of resolving the ideal of fat point subschemes Z=m1p1+...+m7p7 involving 7 general points p1, ..., p7 of P2.

Click here to obtain a plainTeX file of this paper.

7 Fat Points on P2

The algorithm for any fat point subscheme involving 7 fat general points of P2 has been implemented below; to compute the resolution for the ideal of any such fat point subscheme, pick multiplicities of your choice below and click the submit button. (To compute resolutions of ideals of uniform fat point subschemes---i.e., of symbolic powers of ideals of a finite set of general points of P2---click here.)

For the C source code for the program implemented here, computing resolutions for ideals of 7 point fat point subschemes, click here.

mult 1 = mult 2 = mult 3 = mult 4 = mult 5 = mult 6 = mult 7 =

8 Fat Points on P2

Recent joint work of Fitchett, Harbourne and Holay has led to an algorithm for 8 points. Click here for a web based implementation of the algorithm.