\magnification=\magstep1
\baselineskip=14pt
\nopagenumbers
\def\cocoa
{\hbox{\rm C\kern-.13em o\kern-.07
em C\kern-.13em o\kern-.15em A}}
\centerline{\bf COCOA School}
\centerline{\bf Problems on Monomial Ideals}
\centerline{\bf DAY 1: Monday, May 31, 1999}
\vskip .5cm
\item{(1)}
Let $n=6$ and let $\Delta$ be the boundary of an octahedron.
\itemitem{(a)} Determine $I_\Delta$ and $I_{\Delta^\vee}$.
\itemitem{(b)} Compute their respective Hilbert series.
\itemitem{(c)} Compute their minimal free resolutions
\itemitem{(d)} Interpret the Betti numbers obtained in
part (c) in terms of simplicial homology.
\vskip .2cm
\item{(2)}
Explain how {\cocoa} can be used to calculate the homology
of a simplicial complex.
\vskip .2cm
\item{(3)} Consider a $5 \times 5$-matrix of indeterminates
$(x_{ij})$ and let $I_\Delta$ be the ideal in $k[x_{ij}]$ generated by all
$100$ square-free monomials of the form $x_{ij} x_{il}$ or $x_{ij} x_{lj}$.
\itemitem{(a)} The simplicial complex $\Delta$ is called
the {\it chessboard complex}. Explain this name.
\itemitem{(b)} Using Hilbert series in {\cocoa}, find the
number of $2$-dimensional faces of $\Delta$.
\itemitem{(c)} Compute the homology groups of $\Delta$,
using your algorithm in (2).
\itemitem{(d)} Redo the computation in (c) for the finite fields
$k = {\bf Z}_2, {\bf Z}_3, \ldots$. What happens~?
\vskip .2cm
\item{(4)} Let $I$ be the ideal of the cubic Veronese surface
in projective $9$-space. Compute the generic initial ideal of
$I$ for reverse lexicographic order and for purely
lexicographic order. How do their minimal free resolutions
compare to that of $I$ itself~?
\vskip .2cm
\item{(5)} Give an example of a Borel-fixed ideal which is not the
initial monomial ideal of any homogeneous prime ideal in $k[x_1,\ldots,x_n]$.
Are such examples rare or abundent~?
\vskip .2cm
\item{(6)} Let $I \subset {\bf C}[x,y,z]$
be the homogeneous radical ideal of seven
generic points in $\,P_{\bf C}^2$.
\itemitem{(a)} List {\bf all} initial monomial ideals of $I$,
with respect to all term orders.
\itemitem{(b)} How many other Borel-fixed monomial ideals share
the same Hilbert function~?
\itemitem{(c)} Identify the segment ideals in your two lists.
\itemitem{(d)} Do the initial monomial ideals of $I$
all have distinct saturations $I:\langle x,y,z \rangle^\infty$~?
\itemitem{(e)} If we replace the seven generic points by
a point configuration in special position, what happens to the
number of distinct initial monomial ideals~?
\vskip .2cm
\item{(7)}
Let $M$ be an arbitrary monomial ideal in ${\bf C}[x_1,\ldots,x_n]$, and let
${\cal B} \subset {\bf N}^n$ be the set of all vectors $u = (u_1,\ldots,u_n)$
such that $\, x_1^{u_1} \cdots x_n^{u_n} \,$ is not in $M$.
The {\it distraction} of $M$ is the radical ideal $D_M$ of all
polynomials in ${\bf C}[x_1,\ldots,x_n]$ which vanish on the
set ${\cal B}$.
\itemitem{(a)} Determine a finite generating set of $D_M$
\itemitem{(b)} Show that $M$ is the initial monomial ideal of $D_M$
with respect to any term order.
\itemitem{(c)} Determine the prime decomposition of $D_M$.
\itemitem{(c)} The number of prime components of $D_M$ is
called the {\it arithmetic degree} of $M$. Write a {\cocoa} program
for computing the arithmetic degree.
\vfill \eject
\centerline{\bf COCOA School}
\centerline{\bf Problems on Monomial Ideals}
\centerline{\bf DAY 2: Tuesday, June 1, 1999}
\vskip .7cm
\item{(1)} Let $\Delta $ be the simplicial
complex on the set $\{
x_1,x_2,x_3,x_4,
y_1,y_2,y_3,y_4,
z_1,z_2,z_3,z_4 \}$ obtained by {\bf polarization} of the monomial ideal
$\,M = \langle x^4, y^4, z^4,
x^3 y^2 z, x y^3 z^2, x^2 y z^3 \rangle$.
Determine the number of $i$-dimensional faces of $\Delta$
for $i=2,3,4,5,6,7,8$.
\vskip .3cm
\item{(2)}
Let $\prec$ be the purely lexicographic term order.
Using {\cocoa}, compute the generic initial ideal
$gin_\prec(M)$ and its minimal free resolution,
for the ideal $M$ in problem (1).
\vskip .3cm
\item{(3)} Pick $100$ monomials in $x,y,z$ at random with
exponents between $0$ and $1000$. Compute the minimal
free resolution and the Hilbert series of the ideal they generate.
Repeat the experiment ten times. Explain your data.
Try again with more monomials...
\vskip .3cm
\item{(4)} Draw the {\it second barycentric subdivision of a triangle}.
Construct a monomial ideal in $k[x,y,z]$ which has that resolution.
Such ideals exist by Schnyder's Theorem.
\vskip .3cm
\item{(5)} Explain how the Hilbert function command in {\cocoa}
can be used to compute
the Scarf complex of a generic monomial ideal.
Apply your method to compute $\Delta_M$ for
$$ \, M \quad = \quad \langle \,
a^5, \,b^5, \,c^5, \,d^5, \,
a b^2 c^3 d^4,\,
a^2 b^3 c^4 d,\,
a^3 b^4 c d^2,\,
a^4 b c^2 d^3 \,\rangle .$$
The Scarf complex $\Delta_M$ is a
triangulation of the tetrahedron. {\bf Draw it}.
\vskip .3cm
\item{(6)} Compute the irreducible decomposition of
the monomial ideal $M$ in problem (5).
\vskip .3cm
\item{(7)} What is the maximum number of irreducible components
of an artinian ideal generated by $10$ monomials in $4$ variables~?
Can you find an example that attains the bound~?
\vskip .3cm
\item{(8)} Consider the non-generic monomial ideal
$\,M = \langle x,y,z \rangle^3 $. Construct at least three
different free resolutions of $M$ by the technique of
{\bf deformation of exponents}.
\vfill \eject
\centerline{\bf COCOA School}
\centerline{\bf Problems on Monomial Ideals}
\centerline{\bf DAY 3: Wedneday, June 2, 1999}
\vskip .5cm
\item{(1)} Find a monomial ideal $I$ such that
$(I^\wedge)^\wedge \not= I $.
Characterize all of them.
\vskip .2cm
\item{(2)} Show that Alexander duality commutes with
taking radicals:
$$ rad(I^\wedge) \quad = \quad rad(I)^\wedge. $$
\vskip .2cm
\item{(3)} Let $I$ be a monomial ideal given in terms
of its minimal generators, and $a \in {\bf N}^n$
coordinatewise bigger than the exponent vectors
of the generators.
What is the easiest (resp.~fastest)
method in {\cocoa} for calculating the
Alexander dual ideal $I^a$~?
\vskip .2cm
\item{(4)}
The irrelevant ideal of
the product of projective spaces
$\,P^2 \times P^2 \times P^1 \,$
is a square-free monomial ideal $\,M\,$ in
$\,k[ x_0,x_1,x_2, y_0,y_1,y_2, z_0,z_1]$.
\itemitem{(a)} Find the minimal generators of $M$.
\itemitem{(b)} Calculate the minimal free resolution of $M$ in {\cocoa}.
\itemitem{(c)} Interpret the Betti numbers of
your resolution in (b) in terms of convex polytopes.
\itemitem{(d)} Show that the minimal free resolution of
$M$ is a cellular resolution.
\itemitem{(e)} Why is the Alexander dual of $M$ Cohen-Macaulay~?
\vskip .2cm
\item{(5)}
Draw the minimal free resolution of the cogeneric ideal
$$
\langle x^1 ,y^4 ,z^6 \rangle \,\cap \,
\langle x^2 ,y^6 ,z^1 \rangle \,\cap \,
\langle x^3 ,y^3 ,z^3 \rangle \,\cap \,
\langle x^4 ,y^5 ,z^2 \rangle \,\cap \,
\langle x^5 ,y^1 ,z^5 \rangle \,\cap \,
\langle x^6 ,y^2 ,z^4 \rangle. $$
Check your result using {\cocoa}.
\vskip .2cm
\item{(6)}
What is the maximal number of minimal generators of an
intersection of $12$ irreducible monomial ideals in
$\,k[x_1,x_2,x_3,x_4]$~?
\vskip .2cm
\item{(7)}
Show that the hull resolution of
$\, \langle
x^4, y^4,
x^3 z, y^3 z ,
x^2 z^2, y^2 z^2 ,
x z^3, y z^3 \rangle \,$ is minimal.
What is its irreducible decomposition~?
Where is this information hiding in the hull complex~?
\vskip .2cm
\item{(8)}
Compute the hull resolution of the ideal
$$\, I \, = \,
\langle
x_1 x_2,
x_1 x_3,
x_1 x_4,
x_1 x_5,
x_2 x_3,
x_2 x_4,
x_2 x_5,
x_3 x_4,
x_3 x_5,
x_4 x_5 \rangle .$$
Can you state a general result for square-free monomial ideals~?
\vskip .2cm
\item{(9)} Find a codimension $3$ Gorenstein monomial ideal $M$
with seven minimal generators. Show that the
minimal free resolution of $M$
is cellular and given by a convex $7$-gon.
\vfill \eject
\centerline{\bf COCOA School}
\centerline{\bf Problems on Monomial Ideals}
\centerline{\bf DAY 4: Thursday, June 3, 1999}
\vskip .7cm
\item{(1)} Let $M$ denote the monomial module
generated by all Laurent monomials $\,x^i y^j z^k \,$
with the properties that $i+j+k = 0$ and
not all three coordinates of $(i,j,k)$ are even.
Draw a picture of this ideal.
Determine the minimal free resolution of $M$ over $k[x,y,z]$.
\vskip .3cm
\item{(2)} Let ${\cal L}$ be the kernel
of the matrix $\,\pmatrix{3 & 2 & 1 & 0 \cr
0 & 1 & 2 & 3 \cr}$. Show that the hull resolution
of the monomial module $M_{\cal L}$ is minimal.
What happens modulo the action by the lattice ${\cal L}$~?
\vskip .3cm
\item{(3)} Describe the canonical module of the ring $\,k[t^3,t^4,t^5]\,$
as the quotient of a lattice module in
$\,k[x^{\pm 1}, y^{\pm 1}, z^{\pm 1}]\,$ by a lattice action.
Is there a relation to Alexander duality~?
\vskip .3cm
\item{(4)} Using {\cocoa},
compute ${\bf Z}$-graded Hilbert series of
$\,k[t^{20},t^{24},t^{25},t^{31}]\,$ in the form
$$ { p(t) \over
(1- t^{20})
(1- t^{24})
(1- t^{25})
(1- t^{31})}. $$
Give a polyhedral explanation for each term appearing in
the polynomial $p(t)$.
\vskip .3cm
\item{(5)} Suppose you travel to a country
whose currency has four coins valued $20,24,25$ and $31$.
What is the largest amount of money which cannot be
expressed by these coins~?
\vskip .3cm
\item{(6)} Explain how the hull complex of a {\bf generic}
lattice ideal can be computed in {\cocoa}. Apply your procedure
to compute the hull complex for the ideal $I_{\cal L}$ of
the $2$-dimensional sublattice
${\cal L}$ of ${\bf Z}^4$ spanned by the vectors
$\, (-7, -5, 3, 8)\,$ and $\, (4, -7, 9, -1)$.
\vskip .3cm
\item{(7)} Compute the hull resolution for the
ideal of $2 \times 2$-minors of a generic $2 \times 4$-matrix.
\vskip .3cm
\item{(8)} Compute the hull resolution for the
ideal of $2 \times 2$-minors of a generic $3 \times 3$-matrix.
\bye