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.
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.
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.
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.