Determinantal Facet Ideals

We consider ideals generated by general sets of $m$-minors of an $m\times n$-matrix of indeterminates. The generators are identified with the facets of an $(m-1)$-dimensional pure simplicial complex. The ideal generated by the minors corresponding to the facets of such a complex is called a determinantal facet ideal. Given a pure simplicial complex $\mathrm{\Delta }$, we discuss the question when the generating minors of its determinantal facet ideal $J_{\mathrm{\Delta }}$ form a Gr"obner basis and when $J_{\mathrm{\Delta }}$ is a prime ideal.