The Resolving Bracket

In this talk, we define a projective complex and determine those complexes which are generically trivial, that is, which are homologically trivial in general position. A projective complex is basic when the space of first-order syzygies among the points of the configuration is the direct sum of the spaces of firstorder syzygies on the facets of the complex. We define the resolving bracket of a generic basis, a bracket polynomial which vanishes exactly when the generic basis fails, and permits the complex to be lifted to higher dimension. We determine the factorization of resolving brackets, and provide numerous examples. The present report for the conference proceedings is a summary of material to appear (with proofs) in a subsequent publication by the same authors. 1