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.

Using SAGBI basis techniques, we find Gröbner bases for the presentation ideals of the Rees algebra and special fiber ring of a closed determinantal facet ideal. In particular, we show that closed determinantal facet ideals are of fiber type and their special fiber rings are Koszul. Moreover, their Rees algebras and special fiber rings are normal Cohen-Macaulay domains, and have rational singularities.

We introduce the VirtualResolutions package for the computer algebra system Macaulay2. This package has tools to construct, display, and study virtual resolutions for products of projective spaces. The package also has tools for generating curves in ‫ސ‬ 1 × ‫ސ‬ 2 , providing sources of interesting virtual resolutions.

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