\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