We give a complete combinatorial characterization of all possible polarizations of powers of the graded maximal ideal (x 1 , x 2 , . . . , x m ) n of a polynomial ring in m variables. We also give a combinatorial description of the Alexander duals of such polarizations. In the three variable case m = 3 and also in the power two case n = 2 the descriptions are easily visualized and we show that every polarization defines a (shellable) simplicial ball. We conjecture that any polarization of an Artinian monomial ideal defines a simplicial ball. m and making a minimal generatorof J. We call this the standard polarization.

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In this paper, we study different polarizations of powers of the maximal ideal, and polarizations of their related square-free versions.For n = 3, we show that every minimal free cellular resolution of m d comes from a certain polarization of the ideal m d . This result is not true for n = 4. When I is a square-free ideal, we show that the Alexander dual of any polarization of I is a polarization of the Alexander dual ideal of I. We apply this theorem, and study different polarizations of the ideals m d sq.fr and their Alexander duals m n−d+1 sq.fr simultaneously, by giving a combinatorial description corresponding to such polarizations, with a natural dualization. We apply this theory, and study the case of d = 2 and d = n − 1 in more detail. Here, we show that there is a oneto-one correspondence between spanning trees of Kn and the maximal polarizations of these ideals.

In this paper, we study the two natural polarizations, namely the standard polarization and the box polarization, of the d-th power of the maximal ideal in a polynomial ring. We show that these polarizations correspond to smooth points in the Hilbert scheme, and we calculate the dimension of their component which shows that they lie on different components. When d = 2, we show that all maximal polarizations are smooth points, and we give a simple method for calculating the dimension of their component.

We investigate the analogy between squarefree Cohen-Macaulay modules supported on a graph and line bundles on a curve. We prove a Riemann-Roch theorem, we study the Jacobian and gonality of a graph, and we prove Clifford's theorem.