## Fat Point Algorithms

Here are a series of algorithms I've coded over the years and made available to run over the web. Some of the links lead to expository text written some years ago. When you read that something is the "best currently known", keep in mind that "currently" refers to the time at which the text was written. There may have been more recent developments.

Here is a web form for computing actual and expected values of the Hilbert function of the ideal defining n general points of multiplicity m in P2.
Enter the number n of points here:
Enter a single multiplicity m here:

Let a(n,m) denote the least degree of a plane curve which vanishes to order m at each of n general points. Here is a web form for computing various bounds on a(n,m). For more information, click here or here.
Enter the number n of points here:
Enter the uniform multiplicity m at each point here:

Here is a web form for computing bounds on the regularity tind(n,m) of the ideal defining n general points of multiplicity m in P2. For more information, see the subsection titled "Bounding tind(n,m)" of NapConf.html.
Enter the number n >= 3 of points here:
Enter a single multiplicity m >=1 here:

Here is a link to a web based computation of the minimal free resolution of the ideal of n <= 7 general fat points in the plane, with arbitrary assigned multiplicities.

Here is a link to a web based computation of the minimal free resolution of the ideal of n <= 8 general fat points in the plane, with arbitrary assigned multiplicities.

Here is a link to a web based computation of the resolution (or, for n > 9, the resolution conjectured by the Uniform Resolution Conjecture) of the ideal defining n general points in P2 of multiplicity m.

Here is a link to a web based computation of the resolution of the ideal defining any 6 distinct points in P2 of any multiplicities. The points need not be general, and each of the six multiplicities can be assigned arbitrarily. This is based on a joint preprint with E. Guardo.

Here is a link to a web based computation of the resolution of the ideal defining any 6 points in P2 of any multiplicities. The points need not be general, and each of the six multiplicities can be assigned arbitrarily. The points are allowed to be infinitely near, but -KX on the surface X obtained by blowing the points up must be nef. This is based on a joint preprint with E. Guardo.

Here is a link to a web based computation of the Hilbert function of the ideal defining any 7 distinct points in P2 of any multiplicities. The points need not be general, and each of the 7 multiplicities can be assigned arbitrarily. This is based on a preprint joint with A. Geramita and J. Migliore.

Here is a link to a web based computation of the Hilbert function of the ideal defining any 7 distinct points in P2 of any multiplicities, exactly as immediately above, but also gives bounds on the Betti numbers of the minimal free graded resolution of the ideal.

Here is a link to a web based computation of the Hilbert function of the ideal defining any 7 distinct points in P2 of any multiplicities and gives bounds on the Betti numbers of the minimal free graded resolution of the ideal, as immediately above, but uses a better method for giving lower bounds on the Betti numbers.

Here is a link to a web based computation of the Hilbert function of the ideal defining any 8 distinct points in P2 of any multiplicities. The points need not be general, and each of the 8 multiplicities can be assigned arbitrarily. This is based on a preprint joint with A. Geramita and J. Migliore.

Here is a link to a web based computation of the Hilbert function of the ideal defining any 8 distinct points in P2 of any multiplicities, exactly as immediately above, but also gives bounds on the Betti numbers of the minimal free graded resolution of the ideal.

Here is a link to a web based computation of the Hilbert function of the ideal defining any 8 distinct points in P2 of any multiplicities and gives bounds on the Betti numbers of the minimal free graded resolution of the ideal, as immediately above, but uses a better method for giving lower bounds on the Betti numbers.

Here is a link to a web based computation of the Hilbert functions for every set of n <= 8 points in P2 of multiplicity m.

Here is a link to a web based computation of the Hilbert functions for a given set of n <= 8 points in P2 with given multiplicities. It is coded in C, so it will be much faster than the awk version used in some cases above. This version also gives you more control over output.

Here is a link to a web based solution for n < 9 points to a problem posed by Geramita-Migliore-Sabourin.

Here is a link to a web based calculation of the graded Betti numbers for fat ponts in P2 supported at general points. The results assume certain conjectures but in many cases the conjectures are known to hold. Thus the output may only be conjectural but in many cases the output will be the actual graded Betti numbers.

Here is a link to scripts for computing graded betti numbers for points on a plane conic.

Here is a link to a script for finding deficient points on certain secant varieties.

Here is a link to scripts for computing bounds on Hilbert functions and graded Betti numbers for ideals of fat points in P2 with given matroid.

Here is a web page computing Waldschmidt's constant for every choice of up to 8 points of P2.