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\centerline{\bf The toric Hilbert scheme}
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\centerline{Mike Stillman}
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This talk reports on joint work with Irena Peeva.  In this talk,
we motivate and introduce the toric Hilbert scheme, a parameter space
for all ${\cal A}$-graded ideals, for a specific matrix ${\cal A}$.  We
discuss the differences between this scheme and the usual Hilbert
scheme of subschemes of projective space with a given Hilbert polynomial.
We go on to survey what we know about these toric Hilbert schemes.
Finally, we describe the construction of this scheme, and show that it is
the base scheme for a
a universal family of $A$-graded ideals.  One does not need to know much or
indeed
anything about the usual Hilbert scheme in order to follow this lecture.

Let ${\cal A}$ be an $n \times d$ matrix of non-negative integers, and let
$I_{\cal A} \subset S = k[x_1, \ldots, x_n]$ be the corresponding toric ideal.
That is,
  $$I_{\cal A} = < {\bf x}^\alpha - {\bf x}^\beta \mid A(\alpha) = A(\beta)
>,$$
where we use the notation ${\bf x}^v = x_1^{v_1} \ldots x_n^{v_n}$.

The matrix ${\cal A}$ determines a ${\bf Z}^d$ grading of $S$, where the
degree of
$x_i$ is the $i$th column of the matrix ${\bf A}$.  Let ${\bf N}{\cal A}
\subset {\bf N}^d$
be the sub-semigroup generated by the columns of the matrix ${\cal A}$.

An ideal $J \subset S$ is called ${\bf A}$-graded, if it is homogeneous
with respect to the
grading by ${\bf N}{\cal A}$, and it has the same multi-graded Hilbert
function as $I_{\cal A}$.

The {\it ${\cal A}$-toric Hilbert scheme}, or simply {\it toric Hilbert
scheme}, is set theoretically
the set of all ${\cal A}$-graded ideals of $S$.  Denote this set by
$Hilb_{\cal A}$.
We show that this set has the structure of a projective scheme, and that
there is a universal
family, flat over this scheme, such that the fibers are (the zero sets of)
all of the
${\cal A}$-graded ideals.

This scheme has some major differences with the usual Hilbert scheme.
After reviewing quickly
the usual Hilbert scheme in an example,  we describe some of these
differences.

Not much is known yet about these toric Hilbert schemes, other than
some basic facts, the universality property
mentioned above, and some results in low codimension.  As stated above,
these schemes are
all projective schemes.  it is also true that each component is itself a
(possibly non-normal) toric
variety, since one can write down binomial equations defining the toric
Hilbert scheme.
In the case that $n-d = codim(I_{\cal A}) = 2$, much more can be said:
Gasharov-Peeva prove that this scheme is irreducible in this case, and we
prove that it is in fact a smooth toric variety. We mention some open
problems about the structure of these schemes.  In particular, one
significant open question is whether these schemes are connected.  It
appears that an answer would shed light on the Baues conjecture in
combinatorics.

We end the lecture by describing how one constructs these toric Hilbert
schemes, and, if time
permits, we will give a non-trivial example: we have programs written in
the Macaulay 2 language
for finding the local equations of the toric Hilbert scheme, and one can
compute far more examples
than is possible for the usual Hilbert scheme.

We would like to thank Bernd Sturmfels for introducing us to the notion of
parameter
spaces for ${\cal A}$-graded algebras.  In particular, he has a set-theoretic
construction which is reminiscent of our construction.  Our method is
different, in that it gives universality, and therefore a well-defined
scheme structure, and
at the same time, is efficient enough to allow computation of examples to
be made. Bernd also
helped with some of the algorithms which we use to compute the local
equations of the toric
Hilbert schemes.

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